1BDJGJD +PVSOBM PG .BUIFNBUJDT · With this norm the distributions with finite energy form a linear...

PacificJournal ofMathematics

IN THIS ISSUE—Lars V. Ahlfors, Remarks on the Neumann-Poincaré integral equation . . . . . . . . 271Leonard P. Burton, Oscillation theorems for the solutions of linear,

nonhomogeneous, second-order differential systems . . . . . . . . . . . . . . . . . . . . 281Paul Civin, Multiplicative closure and the Walsh functions . . . . . . . . . . . . . . . . . . . 291James Michael Gardner Fell and Alfred Tarski, On algebras whose factor

algebras are Boolean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Paul Joseph Kelly and Lowell J. Paige, Symmetric perpendicularity in Hilbert

geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319G. Kurepa, On a characteristic property of finite sets . . . . . . . . . . . . . . . . . . . . . . . . 323Joseph Lehner, A diophantine property of the Fuchsian groups . . . . . . . . . . . . . . . 327Donald Alan Norton, Groups of orthogonal row-latin squares . . . . . . . . . . . . . . . 335R. S. Phillips, On the generation of semigroups of linear operators . . . . . . . . . . . 343G. Piranian, Uniformly accessible Jordan curves through large sets of relative

harmonic measure zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371C. T. Rajagopal, Note on some Tauberian theorems of O. SzÃ¡sz . . . . . . . . . . . . . 377Halsey Lawrence Royden, Jr., A modification of the Neumann-Poincaré

method for multiply connected regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385George H. Seifert, A third order irregular boundary value problem and the

associated series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395Herbert E. Vaughan, Well-ordered subsets and maximal members of ordered

sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Hans F. Weinberger, An optimum problem in the Weinstein method for

eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413Shigeki Yano, Note on Fourier analysis. XXXI. Cesàro summability of Fourier

series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Vol. 2, No. 3 March, 1952

PACIFIC JOURNAL OF MATHEMATICS

EDITORS

R. M. ROBINSON *R. P. DILWORTH

University of California California Institute of TechnologyBerkeley 4, California Pasadena 4, California

E. F. BECKENBACH, Managing Editor

University of CaliforniaLos Angeles 24, California

*During the absence of Herbert Busemann in 1952.

ASSOCIATE EDITORS

R. P. DILWORTH P. R. HALMOS B0RGE JESSEN J. J. STOKER

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COPYRIGHT 1952 BY PACIFIC JOURNAL OF MATHEMATICS

REMARKS ON THE NEUMANN-POINCARE INTEGRAL EQUATION

LARS V. AHLFORS

1. Introduction. Neumann's method for solving the first and second boundary

value problems lends itself readily to numerical computation. The degree of

success depends primarily on the rate of convergence of the Neumann series, and

the motivation for the present paper arose from a study of this question. The

only new result is an estimate of the convergence rate by means of quasi-con-

formal mappings, which aside from its practical applications seems to present

some theoretical interest. The rest of the paper is concerned with a generali-

zation of the classical approach.

2. The Hubert transform. Let dμ be a complex mass-distribution in the plane

with compact carrier and total mass zero. The energy of the mass distribution is

defined by

\\dμ\\3 = - π ~ l J J Ί o g | ί - T | dμit) d μ ( τ ) .

With this norm the distributions with finite energy form a linear space which can

be completed to a Hubert space S.

The Hubert transform

1

(i) /(*>- — / r2πι ζ- z

defines a function which is analytic outside of the carrier with a double zero at

oo. If the energy is finite one proves that f(z) exists a.e. and

2 / / I / I 2 dxdy- \\dμ\\2 (z =x + iy),Q

where Ω denotes the whole plane.

This result, to which there seems to be no convenient reference, is readily

established by use of Fubini's theorem and the relation

Received September 15, 1951. The paper was written under Contract N5ori-07634between Harvard University and the Office of Naval Research. It was initiated in con-nection with a program of numerical computations of conformal mappings directed byG. Birkhoff.

Pacific J. Math, 3 (1952), 271-280271

272 LARS V. AHLFORS

1 CC dx dy~γ=—- = log Λ - l o g | ί - τ | + 0 ( / Γ a ) .

2 π \z\<R ( ί ~ z ) ^ ~ 2 >

3. The inverse transformation. We regard (1) as a mapping φ which carries

d μ into the differential

fdz = φ(dμ),

considered as element of a Hubert space 3 with the norm

\\fdz\\2 = 2 J J I / I ' dx dy.Ω

The inverse transformation <£~ι exists on a dense subspace of 3. Indeed, let

3 consist of those f dz for which / is continuously differentiable and

ϊL-°ί[tL +i iΔdΊ 2\dz + l dyj

vanishes outside of a compact set. Now 3 is dense in 3, and for / dz C 3 we

can define a mass-distribution with density - 2i df/dT. Its Hubert transform is

1 f(df/dζ dξdη .

π Ω £ ~ z

Integration by parts after removal of a small circle centered at z shows that

this integral has the value f(z).

It follows that φ can be extended to an isometric transformation of S onto

3.If C is a fixed compact set the differentials f dz such that / is analytic out-

side of C form a closed subspace 3 0 of 3. Its image under φ is the closure

£ 0 of the set of mass-distributions over C.

4. Anti-linear mappings. In 8 an involutory anti-linear mapping / is defined

by complex conjugation:

/ dμ = dμ*

A corresponding anti-linear mapping in 3 is defined by

/ ' = Φ J Φ~1

For convenience we shall frequently write

f dz = φ(dμ),

g dz = φidjί).

REMARKS ON THE NEUMANN-POINCARE INTEGRAL EQUATION 273

where gdz is thus the transform of fdz under/'. Clearly,

\\fdz\\ = | | g « * z | | .

We shall also consider

(2) dV = Hfdz - gdz)

and its conjugate differential

dV* = / dz + g dz .

If dμ is real, then / = g, and dV = - 2 A /rfz, dV* = 2 R f dz .

5. Transformations in So and 3 0 . Given a compact set C we now restrict our

attention to the subspaces Co and So introduced at the end of §2. We assume

further that C has zero area, and that the complement of C can be divided in two

open sets B( and S e , the interior and exterior of C*

In these circumstances (2) is more than a formal differential; in δj + Be it

is the differential of the logarithmic potential

V ( z ) = - 7 7 - ' fc l o g \ζ- z \ d μ ( ζ ) .

Moreover, the Dirichlet integral D (V), defined as

Ω

—7χ~

dVdx dy,

can be expressed by

D(V) =Ω

| g | 2 ) dx dy = = \\dμ\\2.

Let us now introduce the linear operator Λ; which transforms f dz into /t dz,

where fi - f for z C δ / and // = 0 for z C δ e , and a similarly defined operator

Λe. Then Λj and Λe are evidently projections on orthogonal subspaces, and

Λ, Λe =

where / is the identity transformation.

Corresponding to Λ/ and Λe we define the operators

Li = φ~ι Λj φt

Le = φ Ae φ

on S o . They are of course also projections, and Li + Le is the identity trans-

274 LARS V. AHLFORS

formation.

The fundamental role in the Neumann-Poincare theory is played by the trans-

formation

T - - [Li - Le + J{Li - Le) / ] .

It operates in So; the corresponding transformation in 3 0 is

S = - [At - Λ e + /'(Λ, - Λ β ) 7']. 1

We note first that JTJ - T; this means that T is a real operator. Secondly,

]LιJ and JLeJ are also projections. For this reason T is self-adjoint with a

norm < 1. We find explicitly

( d μ , T d μ ) = - i [ | | L ^ μ | | 2 + \\Udμ\\2 - \ \ L e d μ \ \ 2 - \\Ledμ\\2].Z

This can also be written in the form

(dμ,Tdμ) = Sj(\fi\2 + \gi\2 - | / e | 2 - | g e | 2 ) dx dy = D ( V( ) - D ( Ve) .

Since

| | r fμ | | 2 - D(F, ) + D(Ve),

we obtain

\(dμ, Tdμ)\ < \\dμ\\2,

and this inequality implies

\\Tdμ\\ < \\dμ\\.

As a matter of fact, the norm of T is given by

\D(Vi) ~ Z ) ( F e ) |(3) | | Γ | | = sup

D{Ve)

6. The Neumann problem. In the classical Neumann-Poincare theory, C is

a smooth curve and dμ a very regular mass-distribution on C. Under these condi-

The operator 5/ ' = j'S is closely related to the operator T introduced by Schifferand Bergman [ l ] . If j'S fdz = Fdz, the function F is equal to the Schiffer-Bergmantransform T/ in Bι.

REMARKS ON THE NEUMANN-POINCARE INTEGRAL EQUATION 2 7 5

tions the Hubert transform

L ζ - z

satisfies the fundamental saltus-condition

where fiiζ) and fe(ζ) are the interior and exterior boundary values on C

For the differentials dV and dV* the corresponding saltus-conditions are

(4) dVi -dVe = 0 ,

dV* - dV* = 2 dμ .

To solve Neumann's problem for the interior means to solve an equation of

the form

(5) dV* = dγ,

where dγ is subject to the condition

Because of (4) the equation (5) can also be written in the form

(6) dμ + — (dV* + dV*) = dγ,

and the corresponding problem for the exterior would lead to the equation

(7) dμ- - (dV* +dV*e) =dγ.

In the general case there is no way of defining dV* and dV* directly. How-

ever, fiiζ) dζ may be interpreted as the saltus of fidz where, as before, /j = /

in the interior and fi = 0 in the exterior. For this reason, we can represent

fiiζ) dζby φ~ιifidz)9 and a similar representation is obtained for feiζ) dζ.

With these interpretations, we find that (1/2) idV* + dV* ) is represented by Tdμ,

and the equations (6), (7) obtain the form

(8) dμ ± Tdμ = dγ .

In this sense the Neumann problem has a meaning under extremely general

circumstances.

276 LARS V. AHLFORS

Equation (8) is solved by Neumann's series

(9) dμ = dγ + Tdγ + T2dγ + ,

provided that the series converges. Here, convergence is interpreted in terms

of the energy-norm, and the necessary and sufficient condition for convergence is

that the norm of T is < 1. More precisely, if | | Γc?μ|| < k | |c?μ| | the series has

a geometric majorant with the ratio k.

It is thus of practical and theoretical importance to find upper bounds for the

norm of T.

7. Quasi-conformality. We shall pursue the study of the convergence only

for the case that C is a Jordan curve, B( and Be being then simply connected on

the sphere 2. It may be expected that the norm of T depends on the regularity and

near-circularity of C, However, instead of introducing explicit regularity condi-

tions, we shall make use of an implicit condition which seems to influence the

norm much more directly.

Our only assumption will be that there exists a quasi-conformal mapping ζ(z)

of Bi onto Be which leaves C fixed. More exactly, ζ(z) is supposed to be a topo-

logical mapping of Bi + C onto Be + C such that ζ( z) = z on C. In B( we assume

ζ(z) to be continuously differentiate, and we write

q dz 2 \dx dy

The Jacobian of the mapping is \p\2 — \q\2; in the present case the mapping is

sense-reversing, and hence \p\ < | g | . The condition of quasi-conformality con-

s i s t s in requiring that

I P I < * Mfor some k < 1. The mapping is then said to have a maximal eccentricity < k

or a maximal dilatation < ( 1 + A ) ( 1 — A).

It can be shown that the existence of a quasi-conformal mapping is compatible

with corners but excludes cusps.

If Bi is a multiply connected region, the harmonic measures of the contours corre-spond to solutions of dμ - Tdμ. In this case the norm is thus exactly one. However, asignificant problem arises if we consider only the sub-space for which the total massvanishes separately on each contour. An even more natural approach is to embed B( in asurface of higher genus so that Be becomes connected [4].

REMARKS ON THE NEUMANN-POINCARE INTEGRAL EQUATION 277

8. Theorem. Our result can now be expressed as follows:

THEOREM. If BI and Be are complementary simply connected regions bound-

ed by a Jordan curve C, and there exists a quasi-con formal mapping of Bi onto Be

which leaves C fixed and whose maximal eccentricity is < k9 then the norm of

the transformation T is at most k.

To make the essential steps of the proof clear, let V be the potential of a

distribution dμ. The restrictions of V to B{ and Be being denoted by Vι and Ve,

as above, we compare V{(z) with Ve (ζ(z)). These functions are equal on C,

and hence

< D{Ve(ζ(z)))

by Dirichlet's principle. But

D(VAζ(z))) = 2 / /dz

dVf

dx dy

dV dVPe w r e

P + — Q dx dy

dζ

dζ

2

2

dVe

dVe

dζ

2\

!

2

( | p I + \q\)2 dx dy

and we have proved that

\q\ - | p |

1 + JcV.) < D(Ve).

J- """ ft

1 - k

Since the inverse mapping of ζ{z) is also quasi-conformal with the same maxi-

mal dilatation, we obtain similarly

These inequalities imply

\D(Vt) - D(Ve)\ < D(Ve)),

and it follows by (3) that the norm of T is < Ic

In this proof the use of Dirichlet's principle needs justification. In other

2 7 8 LARS V. AHLFORS

words, we must prove that the mixed Dirichlet integral

D(V.(z) - Ve(ζ(z)), ? . (*))

vanishes . For this purpose it is sufficient to know that the potential V(z) is

continuous on C. Then V{ (z) - Ve (ζ{ z)) i s continuously zero on the boundary.

Moreover, we have shown that Ve (ζ(z)) h a s a finite Dirichlet integral. Under

these conditions the mixed Dirichlet integral vanishes according to a familiar

theorem.

We have thus to show that V{ z) i s continuous for a dense se t of distributions

in 8 0 . Suppose that w = φ(z) defines a conformal mapping of Be onto \w\ > 1.

Set

e(10) w = φ(z) for zCBe

w = \/φ{ζ{z)) for z CBr

This is a quasi-conformal mapping of the whole z-plane onto the w-plane except

for C where the mapping is only known to be continuous. It is an important theo-

rem in the theory of quasi-conformal mappings that such a mapping satisfies a

Lipschitz condition of order (1 — &)/( 1 + A) on any compact set 3. We have thus

\ φ ( Z ι ) - φ { z 2 ) \ < M \ z r -

in a neighborhood of C.

From this behavior we draw two conclusions: 1) if a mass-distribution dμ of

finite energy is carried from the z-plane to the w-plane by means of the mapping

(10), then the energy remains finite, and vice versa; 2) if the potential is con-

tinuous in one plane, then it is also continuous in the other.

In the u -plane the distributions with continuous density in | w \ = 1 are dense

and have a continuous potential. The corresponding distributions in the z-plane

have the same property. This completes the proof.

9. An application. As an application, we consider the case of a star-shaped

region B( whose boundary has the equation r = r(φ) in polar coordinates. A

3 Assume that ζ(z) maps |z| < 1 quasi-conformally onto | £ | < 1 with dilatation < K =(1 + k)/(l — &). The mapping can be extended to the double discs with branch-points atz 1 > z 2

a n d £( z i)> £ ( z 2 ) Conformally speaking the double discs can be regarded as annuli,and the existence of the mapping implies that the ratio of their modules must be < K. Asz 1 and z2 approach each other the modules are asymptotic to — log | z± — z2\ and~ l°g l £ ( z i ) "~ ζ(z2 )|» an<* l ^ e Lipschitz condition follows. The proof is inspired byreasonings of Teichmuller.

REMARKS ON THE NEUMANN-POINCARE INTEGRAL EQUATION 2 7 9

quasi-conformal mapping ζ{z) is obtained as a transformation by reciprocal radii:

Z

Since

dφ 1 dφ 1

σz 2ιz dz 2iz

we obtain

<?£ r 2 rr'q = —r = - — + ί — .

dz z2 pHence

lpl \r'\ a= . = cos β9

\q\ y r

2 + r'2

where Θ is the angle between the tangent and the radius vector. If θ0 = min θ the

Neumann series has thus a convergence ratio < cos #0 .

10. Rate of convergence* For practical purposes it may not be sufficient to

know that the series (9) converges in the energy norm. If C is rectifiable we con-

sider the Banach space B of distributions dμ with continuous density on C (and

total mass zero) and use the maximum of the density as norm. Under sufficiently

strong regularity conditions it is proved in the classical theory that T maps B

upon itself, and even that T is completely continuous on B.

If this is so, T has a pure point-spectrum in B. But it is clear that every

eigen-value in B belongs to the spectrum with respect to Eo. In Eo the absolute

value lλo( of the spectral point nearest to the origin is reciprocal to the norm.

The smallest eigen-value in B is > | λ0 |, and it follows that the Neumann series

dγ + XTdγ + X2T2dγ + •••

converges in the B-norm for | λ | < | λ0 |. Therefore the convergence rate of

dγ + Tdγ + T2dγ + •••

is comparable to that of a geometric series with ratio | λ0 I""1.

An upper bound for the norm of T in So can therefore serve as a practical esti-

of the rate of convergence in solving a Neumann problem by the iterative method.

280 LARS V. AHLFORS

REFERENCES

1. S. Bergmann and M. Schiffer, Kernel functions and conformal mapping, CompositioMath. 8 (1951), 205-249.

2. G. Birkhoff, D. M. Young, and E. H. Zarantonello, Numerical methods in conformalmapping, to appear in Proc. Symposia Appl. Math. 4 .

3. S. A. Gerschgorin, Uber die konforme Λbbildung eines einfach zusammenhangendenBereiches auf den Kreis, Mat. Sbornik 40 (1933), 48-58.

4. H. L. Royden, A modification of the Neumann-P oincare method for multiply con-nected regions, Pacific J. Math. 2 (1952), 385-394.

HARVARD UNIVERSITY

OSCILLATION THEOREMS FOR THE SOLUTIONS OF LINEAR,NONHOMOGENEOUS, SECOND-ORDER DIFFERENTIAL SYSTEMS

LEONARD P. BURTON

1. Introduction. Oscillation theorems for the solutions of the equation

ax

are classical. It is the purpose of this paper to develop theorems of a similar

nature for a class of equations of the type

d \ dyλ— \K(x)—\ - G(x)y = A(x).ax L ax 1

It will be assumed that over an interval X: a £ x £ b (b > α), the functions

K(x), G(x)9 and A(x) are continuous. All quantities used are assumed to be

real. Primes will be used to indicate derivatives with respect to x.

Use will be made of the following lemma which gives a modified form of

properties of the second-order linear homogeneous equation developed by W. M.

Whyburn[3, pp.633-634].

LEMMA 1. Lety(x), a solution of {Ky')'- Gy = 0 over X, have the m zerosΓi> '•" > rm ( m > 2) on X. Let the inequalities K > 0, G < 0 hold, and let GK

be a nonincreasing function of x on X, If A is nonvanishing except possibly at

α, and for x > a either one of the following is true over X:

(a) A > 0 and A/G is a strictly decreasing function of x9

(b) A < 0 and A/G is a strictly increasing function of x,

then

A ( t ) y ( t ) d t \ < \fΓί + 2 A ( t ) y ( t ) d t \

Received January 16, 1952.Pacific J. Math. 2 (1952), 281-289

281

282 LEONARD P. BURTON

In order to prove this lemma one needs only to make straightforward modi-

fications in the arguments given by Whyburn.

LEMMA 2. Under the hypotheses of Lemma 1, the zeros of

x A(t)y(t)dt

and y(x) separate each other on a < x <_ b.

This result, which also was given by Whyburn, is an immediate consequence

of Lemma 1.

LEMMA 3 . Let u(x) be any solution of the system (Ky')' - Gy == 0 , y ( b ) = 0 .

Under the hypotheses of Lemma 1, J A(t)u(t)dt does not vanish in a £ x <

b.

Proof. If u(x) has no zero except b on X, the conclusion is obvious. Other-

wise, by Lemma 1, if q is the last zero of u(x) on X preceding b, then the

integral/ A (t)u (t )dt has the sign of f A(t)u(t)dt.

For the sake of brevity we shall henceforth let (H) represent the following

set of conditions on X.

(1) K(x) > 0, G(x) < 0.

(2) K(x)G(x) is a nonincreasing function of x

(3) Either one of the following is true:

(H)\ ( i ) β < 0, A(x) > 0 for x > a and A(x)/G(x) is a strictly de-

creasing function of x

( i i ) β 2l 0, A(x) < 0 for x > a and A(x)/G(x) is a strictly in-

creasing function of x.

Let Uγ(x) be any solution of (Ky')'— Gy = 0 such that ux(b) = 0 Choose

another solution u2(x) such that K(u2u[ — u'2ux) = 1 on X. As a final pre-

liminary result we have the following:

LEMMA 4. Under the hypotheses (H) if β φ 0, then

— £ — fb A(t)Uι(t)dt> 0u2(b) x

over a < x < b.

Proof. By Lemma 3, J A(t)uι(t)dt has the same sign over a <_ x < b as

SECOND-ORDER DIFFERENTIAL SYSTEMS 2 8 3

J A(t)uί(t)dt9 where q, is the la s t zero of ux (x) on a _< x < b (or where

q; = a if uί(x) h a s no such zero). From K(u2u[ - u2ux ) = 1 we obtain \/u2 (&) =

K ί . Hence

—^77 Jh A(t)Uι(t)it=K(b) fb [ j84(ι) lK(*) i* i(O]ώu2(b) If <lf

and this latter expression is positive since the integrand is the product of two

negative quantities.

Hereafter free use will be made of the facts that any solution of (Ky'Y -

Gy — 0 can have only a finite number of zeros on X and that, under the hypothesis

GK < 0, the zeros of any two linearly independent solutions separate each other.

2. Oscillation theorems. Let yι(x) be any solution of (Ky')' ~ Gy = A

over X which satisfies the condition y(^) = jS. Then y\(x) can be expressed

in the form

yι(x)= c u ^ * ) * — — - u2(x) + uγ(x) Jx A(t)u2(t)dtu2(b) a

u2(x) f A(t)Uι(t)dt,

where ^(x) and u2{x) are as in Lemma 4, and c is a constant. We shall prove

the following result.

THEOREM 1. Under the hypotheses (H) the zeros of yί(x) and uι(x) sepa-

rate each other on a < x < b.

[If β £ 0 the restriction that A/G be strictly increasing or decreasing may

be modified to the extent of allowing A/G to be a monotone increasing or de-

creasing function. Under the modified hypotheses it can be shown that

— £ - fb A(t)Uι(t)dt > 0 ,u2(b) x

and since β/u? (b) is not zero the proof of the theorem is still valid.]

Proof. The functions y\(x) and ιiι(x) cannot vanish simultaneously on X

except at b; for, letting q be a zero of y\{x) and ux(x) one obtains

= u2(q) \f* A(t)Uι(t)dt + j S / i i a ί i ) ! = 0 .

This is impossible since u2(q) φ 0 and, by Lemmas 3 (if β = 0) and 4 (if

β Φ 0), the expression in brackets never vanishes.

Suppose now that q and q' Φ b, (q < q')y are consecutive zeros of ux(x),

and that yι(x) does not vanish at any point of q < x < q'. Then, by Rollers

Theorem, [uί(x)/yί(x)Y must vanish at least once in this interval. But

fb A(t)Uι(t)dt+ β/[u2(b)]

and, as above, this expression never vanishes.

In a similar manner it can be shown that between two consecutive zeros of

Ύι{x)9 ui(x) must vanish at least once.

COROLLARY 1. If β Φ 0, the zeros ofγί(x) and ux(x) separate each other

on a < x < b.

Proof. If β ?4 0, the above argument is valid with q' - b.

COROLLARY 2. // ux(x) has m zeros on X, then Ύι(x) has either m - 1,

m, or m + 1 zeros on X.

Proof. Let q0 be the first zero of ux(x) on X and qf be the last zero of

u±(x) preceding b; Ύι(x) may or may not have a zero in a £ x < qQ. In the

interval q0 < x < q^ Ύ\(x) has exactly m - 2 zeros. If β Φ 0, then y{(x) has

exactly one zero in qr < x < b by Corollary 1. If β = 0, Ύι(x) may or may not

vanish in qr < x < b. (See Theorem 4.)

The next theorem is applicable only if the system (Ky')' - Gy = 0, y ( α ) =

y ( ό ) = O is incompatible. In this case ( i ) one can select linearly independent

solutions uι(x) and u2(x) of (Ky')' — Gy = 0 such that u ι ( i ) = u 2 (α) = 0 and

K(u2u[ — u2ui) = 1 on ί and ( i i ) the nonhomogeneous system (Ky')' — Gy = Af

y ( α ) = 0, y(b)- β has a solution, say y 2(*)* ^ e t n e n ^ a v e ι ^ e following result.

THEOREM 2. Lei ίAe hypotheses (H) be satisfied. Assume that u2(x)

oscillates on X, and let a = pl9 p2, ••• , pm (m > 3) be its consecutive zeros.

Then, for i 4 1> Ύ2(Pi) ^ 0 an^ either y2(x) has two zeros in (pj, Pι + 1 ) α ^

T&orce in (p ι + 1 , Pj + 2 ) ^ ~ ι — /n — 2 ) , or vice versa. In the interval a < x < p2,

y 2 ( # ) Aα5 either no zero or one zero. In the former case it has two zeros in

(p 2 , p3 ), in the latter case it has no zero in (p 2 , p3 ). // y2(x) has two zeros

in ( p m - 1 , pm ), it has no zero in pm < x < b.

Proof. The function y2(x) can be expressed in the form

SECOND-ORDER DIFFERENTIAL SYSTEMS 285

2(χ) + "ι(*) C A(t)u2(t)db + U2(x) f A(t)ux(t)dt.a χf

χ

If 72 (χ) n a s three or more zeros in (pj, p ι + 1 ) ( 2 £ ί £ m — 1), Theorem 1

requires that uι(x) have more than one zero in that interval; this is impossible

since the zeros of ut(x) and u2(x) separate each other. Also, y2(x) cannot

have a single zero in (p;, P ί + 1 ) (2 £ i £ m - 1), for then y2(pi )y2(p ί + 1 ) < 0

and such a product is always positive. To see this, notice that

y 2 ( p i ) = W i ( P f ) / l A ( t ) u 2 ( t ) d t = u t ( p i ) F ( p i ) 9

where F (x) = y .4 ( ί ) u 2 ( ί ) </ί as in Lemma 2. Since the zeros of both uί(x)

and F(x) separate those of u2(x), the product uί{pι) F (pi) (2 £ i £ m) is

consistently positive or negative. Thus y2(pi ) 7 2 ( P t + i ) > 0 ( 2 £ ί £ t f i — 1) .

The function ux{x) has a zero in each of (p, , P ι + 1 ) and (p ί + 1> P,+ 2 ^ ^ —

ΐ £ Hi — 2 ) . By Theorem 1, y 2 ( * ) m u s l ; have a zero in (p;, P i + 2 ) If 72(^)

has no zero in (pi, Pι + ι)» it must have one, and therefore two, in ( p ι + 1> Pj + 2 )

Now assume that y2(x) has two zeros in (p t , p ι + 1 ) H y2^χ) a^so ^ a s t w o zeros

^n ^P* + i ' Pi + 2^' ^ e n u i ^ % ^ must have three zeros in (p t , Pj+2)? t u t this is

impossible. Hence y 2 ( # ) h a s no zero in (p ι + ι> Pj+ 2 )

This same type of argument can be used to prove the part of the theorem

pertaining to the interval a < x < p2 and the interval pm < x < 6.

REMARK. Theorems 1 and 2 are not true in case β ^ 0 without the re-

striction βA(x) < 0, x > α . This is shown by the example

ί— y'\ y ( 0 ) = 0, y(\/"^Γ) = -9π.

Here βA(x) = 9πx3 > 0 on 0 < x < \J9π. The solution of the given system is

y(%) = -% 2 , which does not oscil late. However, each of ux(x) = - cos (x2/2)9

u2(x) = sin (x2/2) has five zeros on 0 £ x £ \f 9π.

3. Application to a system involving a parameter. It will now be supposed

that K, G, and A are continuous functions of (x, λ) when a £ x £ b, Ax <

λ < Λ2. The system

[K{x, λ)y'Y - G(x, λ)y = 0, y(a, λ) = 0, y(b9 λ) = 0,

is a system of Sturmian type. Let K and G satisfy conditions sufficient to assure

the validity of known oscillation theorems for this system [ 1 , p. 66] to the

extent that there exists an infinite set of characteristic numbers λi, Λx < ΛQ <

• < λm < < Λ2, having no limit point except Λ2, and such that if um is the

characteristic function corresponding to then υ^ has TO zeros in a < x < b.

Let v2(x, λ) be the solution of

[K(x,λ)y'Y - G(x,λ)y = 0

satisfying the initial conditions v2(a, λ) = 0, v2(a, λ) = σ, where σ is a posi-

tive constant. By the fundamental existence theorem [ l , p. 7], v2(x, λ) is a

continuous function of x and λ. It is well known [2, pp. 229, 232] that as λ in-

creases from Λt a new zero of v2(x, λ) appears at b for λ = λj ( i = 0, 1, •)>

and that each such zero moves continuously towards a as λ increases continu-

ously.

For each λ, let vι(x9 λ) be a solution of

[K(x,λ)y'Y - G(r,λ)y =0

satisfying the condition vx{b, λ) = 0. If λ = λ; ( i = 0, 1, •)> v^x, λ) is simply

a constant multiple of v2(x, λ) . For λ ^ λj, vx(x, λ) and i>2(#, λ) are linearly

independent. It follows that on X, for λ < λ 0 , vx(x, λ) has a zero only at b;

for λm £ λ < λ m +i (m = 0, 1, •••)> ^ι(*> λ) has m + 2 zeros. Theorem 1 and

its corollaries apply to give the following result.

THEOREM 3. Let the system

[K(x,λ)y'Y - G(x,λ)y = A(x,λ), y(b, λ) = β(λ),

for each fixed λ in (Ai9 Λ2 ), satisfy the hypotheses (//). Let yχ(x, λ) be a

solution. Over X: a < x _< b, if

β(λ) Φ 0 and λ < λo, then y$x, λ) has either no zero or one zero,

λ = λm (m :> 0), then yx(x, λ ) has m + 1 zeros,

λm < λ < λ m + 1 (m >, 0), ίAerc yι(x, λ) Λαs m + 1

or m + 2 zeros;

β($ = 0 αncί λ < AQ, then yι(x9 λ) Aαs either one zero or two zeros,

λ = λm (TO >_ 0), then yχ{x, λ) A«5 m + 1 or m + 2 zeros,

λm < λ < λ m + ! (TO 2l 0) then yt(x9 λ ) has m + 1, TO + 2,

or TO + 3 zeros.

COROLLARY. AS λ increases in (Λ l f Λ2 ) the number of zeros of the so-

lutions jγ (x, λ) increases indefinitely.

Interesting and more precise results can be obtained in connection with the

two-point system

[K(x,λ)y'Y- G(x,k)y = A{x,\),(5 λ)

y(a,λ) = 0, y(δ,λ) = 0,

where K, G, A conform to the hypotheses (H) for each fixed λ in (Λ^ Λ2 ). If

λ = λ, (f= 0, 1, )f then (Sy) is of course incompatible. Otherwise, for each

λ one can choose υx {x, λ) such that

Vi(b, λ) = 0, v[(b, λ)K(b,k)v2(b,λ) '

so that

K(x,λ)[v2(x,λ)vί(x,λ) - vί(x9λ)vx(x,λ)]

^ K(b9\)[v2(b,λ)υί(b,λ)]= 1

on X. It follows that vι(a, λ) = - l/[σK(a, λ)] is negative for all λ.

The solution of (S^) can be expressed as

y2(*.λ) = vi(*,λ) fa

x A{t9λ)v2(t,λ)dt

+ va(*.λ) fχ

b A(t, λ)vι(t,λ)dt.

We now consider an interval Lm: λm < λ < λm+i, and let Xo represent the

interval a < x < b. For a fixed λ in L m , each of vx{xy λ) and v2(x, λ) has

m+ 1 zeros on >Y0 For the sake of definiteness let A(x, λ) be positive over

X0Lm; and let m be odd so that, for λ in Lm, vt(x, λ) and v2(x, λ) each has

an even number, m + 1, of zeros in Xo. Then f[ (ό, λ) > 0 and, by virtue of

Lemma 3,

y ί ( α , λ ) = σ fb A{t,λ)vι(t,λ)dt

is negative. Let qΛλ) represent the last zero of vι(x, λ) preceding b. By The-

orem 1 it follows that y2[qf(λ), λ] is positive over Lm. However, y2 (6, λ) is

2 8 8 LEONARD P. BURTON

negative for λ sufficiently close to λm, positive for λ sufficiently close to

because

y 2 ' (*»λ)= υ{{b,λ) fa

b A(t,k)v2(t,λ)dt,

and by Lemma 3

fa A(t, λm)v2(t, λm)dt < 0 and ^ A (t, λm+ι)v2(t, λm+ t) dt > 0.

Since y2(x, λ) is a continuous function of (x, λ) over XLm [l, p. 114], it follows

that there exist €m > 0 and 6 m + ι > 0 such that for λ' - λm < €m and λm + t -

λ" < £m + i> 72( > λ') has no zero in qλλ') < x < b, and y2(xy λ") has one

zero in qΛλ") < x < b.

A similar argument can be made in case m is even or in case A(x, λ) is

negative over /to This proves the following result.

THEOREM 4. £e£ (Sλ) satisfy the hypotheses (H) for each λ in (Al9 Λ2 ).

On Xθ9 y2(x, λ) has m zeros for λ sufficiently close to λm, /n + 1 zeros for λ

sufficiently close to λm+ι (m = 0, 1, 2, ) •

Letting po(λ) be the first zero of v2(xr\) to the right of α, one readily

sees that

y 2 [p 0 (λ) , λ] = vt[Po(λ)9 λ] j[ P o λ ) ^ ( ί , λ ) t ; 2 ( ί , λ)Λ

is positive or negative according as A is positive or negative. If A > 0, then

y2{a, λ) = α J /4 (ί, λ) vγ{t, λ) ώ is positive or negative over Lm according

as m is even or odd. If A < 0, y2 (o, λ) is negative or positive over Lm accord-

ing as m is even or odd. If one uses these relations as well as Theorem 1 and

Theorem 2 to sketch graphically several typical cases, he obtains a striking

illustration of the effect of the discontinuities of the function y2^χy λ) at the

characteristic values of λ. Finally, one may observe that, regardless of the

sign of A, for an even value of m the first zero of v2(x, λ) on a < x < b pre-

cedes the first zero of y2(x, λ), and for an odd value of m the opposite is the

case.

REFERENCES

1. M. Bocher, Lecons sur les methodes de Sturm, Paris, 1917.


2. E. L. Ince, Ordinary differential equations, Dover, New York, 1926.

3. W. M. Whyburn, Second-order differential systems with integral and k-point bounda-ry conditions, Trans. Amer. Math. Soc. 30(1928), 630-640.

UNIVERSITY OF CALIFORNIA

DAVIS, CALIFORNIA

MULTIPLICATIVE CLOSURE AND THE WALSH FUNCTIONS

PAUL CIVIN

l Introduction. In his memoir [l] On the Walsh functions, N. J. Fine util-

ized the algebraic character of the Walsh functions by interpreting them as the

group characters of the group of all infinite sequences of O's and l 's, the group

operation being addition mod 2 of corresponding elements. In this note, we in-

vestigate some properties of real orthogonal systems which are multiplicatively

closed. We show that any infinite system of the stated type is isomorphic with

the group of the Walsh functions. Furthermore, under hypotheses stated in Theo-

rem 3, there is a measurable transformation of the interval 0 ^ x < 1 into

itself, which carries the Walsh functions into the given system of functions.

As is well known, the Walsh functions are linear combinations of the Haar

functions. B. R. Gelbaum [2] gave a characterization of the latter functions in

which the norm of certain projection operators and the linear closure in L of

the set of functions played an essential role. Although the characteristic fea-

tures of the Walsh functions are, aside from orthogonality, totally distinct from

those for the Haar functions, there is some similarity of proof technique in

establishing the characterization.

For the sake of completeness, we define the Rademacher functions {φn{x)\

as follows:

(1.1) φo(x)1 (0 < x < 1/2)

φo(χ + 1) = φo(χ).-1 (1/2 < x < 1)

(1.2) φn(x) = φo(2"n) (n = 1, 2, . . . ) .

The Walsh functions, as ordered by Paley [3], are then given by

(1.3) φo(x) = 1,

and

(1.4) if n = 2Uχ + 2n<2 + + 2Uk (nγ < n2 < < τ% ) , then

Received September 21, 1951.Pacific J. Math. 2(1952), 291-295

291

292 PAUL CIVIN

Throughout we denote the Lebesgue measure of a set E by μ(E). Equalities

of functions are in the sense of almost everywhere equalities, and considerations

of sets should be interpreted "modulo sets of measure zero".

2. Algebraic correspondence. Let us consider any sequence {An(x)} of

functions satisfying the following conditions:

(2.1) λn{x) is measurable, real valued, and

λn(x + 1) = λnU) in = 0, 1, ••• ),

(2.2) jΓ l λn(x)λm(x) dx= S™.

(2.3) The sequence \λn(x)} is closed under multiplication.

THEOREM 1. Any system of functions satisfying (2.1)-(2.3) is a commuta-tive multiplicative group.

Proof. Associativity and commutativity are immediate. Let if j be given

nonnegative integers, and let k9 m, and n be defined by the formulas

λj(*) λ/(*) = \k(x), λ?(*) = λ m ( * ) , λj{x) = λB(ίc) .

Then

1= jf1 λ\(x)dx- fo

l λm(x) \n(x) dx .

Hence m = n, and by a renumbering we have λ?(x) = λo(x), for all i. In par-

ticular, λ^(x) = λo(x), and for almost every x either λo(*O = O or λ o ( * ) = l .

Since

y* λ*(x) dx = 1, we have λ o (^) = 1 almost everywhere.

Thus the set \λn(x)\ possesses a multiplicative unit; and since A?(%) = 1

for all i, the set iβ a group.

THEOREM 2. The multiplicative group A with elements λn(x) is isomorphic

MULTIPLICATIVE CLOSURE AND THE WALSH FUNCTIONS 2 9 3

with the multiplicative group Ψ with elements the Walsh functions, ψn(x)

Proof. We first reorder the sequence \λn{x)} Let v0 (x) = λ o ( % ) a n d

ι/ί (x) = λ ^ x ) . For k - 1, 2, , define v2k (x) a s the first se t in the sequence

lλn(x)] which follows v2k-\(x) in that sequence, and which is not a product

of elements of the sequence \λn(x)\ which preceed it in that sequence. For

n = 2 * ι + . . + 2* Γ , kx < k2 < - . . < kΓ9 let vn(x) = Π v kj(x).

By virtue of (2.3), each vn(x) is a λj{x) Also each λj(x) is a vn(x), by

the method of construction, and the representation is easily seen to be unique.

Thus the sequence \vn{x)\ is a rearrangement of the sequence \\n(x)\ The

isomorphism between the groups Λ and Ψ is now evident upon pairing vn(x) and

We have incidentally established the result that any denumerable group all

of whose elements satisfy a2 = 1 is isomorphic to Ψ.

3. Analytic transformation. We introduce the terminology that the subscripts

ht hf *•' * h a r e unrelated if no λιk(x) is a product of the remaining λj (%).

In the proof of Theorem 1, we incidentally established the result that λn(x)

takes on only the values + 1. Let

(3.1) Pk = ix: λk(χ) = 1 ; 0 < x < l } ,

(3.2) Nk = {x: λk{x) = - 1 ; 0 < Λ < U A = 1, 2, ••• .

It is convenient to adopt the convention that Ek and Fk represent some ordering

of the pair P^ and Λ^

LEMMA 1. lfi\f 12, ••• , ir a r e unrelated, then

Proof. Let

r

g(x) = 2 ~r

where

294 P A U L C I V I N

δ ; = 1 if Ei. = Pi},

and

δ ; = - l if Ei. = Nir

Then g(x) is the characteristic function of ΓK E{.. Hence

( i + δ, λίy iχ))dx

2- / ' 1 + Σ 8/ λl;. + Σ δj δj* λij(x) λ i y , ( * ) + • dx = 2"

since the integral of any product of λ's with unrelated indices is zero.

COROLLARY. Any infinite product of sets E(, without repetitions, has

measure zero.

Proof. Of the subscripts involved in the infinite product, an infinite number

must be unrelated. The product set is thus a subset of a set of measure 2""Γ for

each r.

In the remainder of the discussion it is convenient to revise the notation

so that P&, Nfc, Ek are associated with v2k(x) rather than with λ& (x)

THEOREM 3. If any system of functions satisfy (2.1)-(2.3) and for almost

every choice of E^ as Pjς or Nfc, the set Π^= £& consists of a single point, then

there is a transformation T, defined almost everywhere, of the interval 0 _£ x < 1

into itself with the following properties:

( i ) T, where defined, is one-to-one except possibly over a denumerable

set over which it is two-to-one;

( i i ) vk(x) = ψk(Tx);

(i i i ) T is a measurable transformation;

(iv) For any measurable set E, μ(T~ι E) - μ ( £ ) .

Proof. The hypothesis of the theorem implies that almost every x, 0 < x < 1,

is in exactly one set Π^=o E^m To any such x, x ^L Π~=o Λ , associate the

number Tx according to the following scheme. Let the Ath position of the dyadic

expansion of Tx be 0 if Ejc-ι = Pk-\ and 1 if £&-i = Nk-ι ^ e thus obtain a

transformation, defined almost everywhere, of the interval 0 < x < 1 into itself.

MULTIPLICATIVE CLOSURE AND THE WALSH FUNCTIONS 2 9 5

Whenever defined, the transformation is one-to-one except possibly on the in-

verse image set of the nonzero dyadic rationale where it is exactly two-to-one.

By virtue of Theorem 2, all we need do to establish ( i i) is to show that

v k {x) = ψ k ( Tx) (k = 0, 1, ). For x C Pk, Tx has the (k + 1 )th place

of its dyadic expansion 0. Hence

Similarly, if x C N^, then

= - 1

In the interval 0 _ x < 1, any dyadic interval consists of numbers whose

dyadic expansions have their first r digits α t ar and the remaining digits

arbitrary. The inverse image of such an interval is the set ΠΓ~j£y, where Ej = Py

if αy+i = 0 and Ej = /Vy if αy + i = 1. Lemma 1 shows that the measure of the in-

verse image is 2~Γ. Hence for dyadic intervals, the inverse image is a measur-

able set of the same measure. Since any open set is the union of disjoint dyadic

intervals, the same statement can be made for any open set, and similarly for

any closed, Fσ, or Gg set. A classical argument then shows the statement to be

valid for any measurable set £•

REFERENCES

1. N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65(1949), 372-414.

2. B. R. Gelbaum, On the functions of Haar, Ann. of Math. 51(1950), 26-36.

3. R. E. A. C. Paley, A remarkable class of orthogonal functions, Proc. London Math.Soc. 34(1932), 241-279.

UNIVERSITY OF OREGON

ON ALGEBRAS WHOSE FACTOR ALGEBRAS ARE BOOLEAN

J. M. G. FELL AND ALFRED TARSKI

1. Introduction.1 We consider an algebraic system 2Ϊ = {A, +) constituted

by an arbitrary set A and a binary operation +. The set A is assumed to be

closed under +, and to contain a (uniquely determined) zero element, that is, an

element 0 such that

for every x in A. We shall refer to such a system simply as an algebra. By a

subalgebra B of ?I we understand an arbitrary subset of A which is closed under

+ and contains 0 as an element.

For two subalgebras B and C, a function /, whose domain includes (but does

not necessarily coincide with) B and which maps B onto a subset of C in such

a way that

for all bl9 b2 in B, is called as usual a (B, C) -homomorphism; if in addition / i s

biunique on B, and maps B onto the whole of C, it is called a (B, C)-iso-

morphism.

A relation R holding between certain pairs of elements of A is called a

congruence relation over 21 if (i) R is an equivalence relation whose field is

A, and (ii) (α + b) /?(α' + b') whenever α/?α'and bRb'; here we have expressed

symbolically by cRd the statement that R holds between the elements c and d.

Suppose that R is a congruence relation over 21. For each element a of A, the

coset of a under R, in symbols a/R, will be defined as the set of all b in A for

which aRb. It is easy to see that any two cosets are either identical or else

have no element in common, and that the set-theoretical union of all the cosets

is simply A. If a/R and b/R are any two cosets, we denote by a/R +'b/R the

coset (α + b)/R. By condition (ii) of the definition of a congruence relation,

this determines an operation + ' on pairs of cosets. The algebra consisting of

*A detailed discussion of all the notions and results contained in the Introductionwill be found in [8] (see in particular Appendix § § A and B) and [4] (see in particular§ § 1 and 2).

Received July 3, 1951.Pacific J. Math. 2 (1952), 297-318

297

2 9 8 J. M. G. FELL AND ALFRED TARSKI

the family of all cosets of elements of A under R9 with the binary operation +',

is called the coset algebra (of 21 under J?) and is denoted by 21/7?. 2

Given a subalgebra B of 21, by the center of β—in symbols β c - w e mean the

set of all c in B satisfying the following conditions: (i) there is an element d

in B with c + d = 0; (ii) the commutative-associative formula

(bx + b2) + c = bt + (b2 + c) = (bι + c) + b2

holds for any bl9 b2 in B. 3 Clearly 0 belongs to the center of all subalgebras.

It is not hard to see that b + c = c + b whenever b £ B and c £ β c (commutative

law); and that b± + c = b2 + c implies bγ = b2 for all bu b2 in B and c in Bc

(cancellation law). It follows that for any subalgebra β, the system ( β c , +) is

an Abelian group. A subalgebra B in which Bc = {0}, that is, Bc consists only

of 0, is called centerless. Similarly, an algebra 21 = {A, +> in which Ac = 1 0 } is

called centerless.

A subalgebra D is called the direct product of subalgebras B and C if (i)

every element d in D can be uniquely represented in the form d = b + c, where

b £ β and c £ C; (ii) we have ί> + c £ D for every 6 in β and c in C; (iii) the

formula

(£>! + c t ) + (b2 + c2) = (&! + ί>2) + ( q + c2)

holds for arbitrary bi9 b2 in β, and cl9 c2 in C The subalgebra D with these

properties (if it exists at all) is of course uniquely determined by B and C, and

will be denoted by β x C It is clear that if B x C exists, and β λ and Ct are

subalgebras included respectively in B and C, then β χ x Cx exists. It can be

shown without difficulty that the operation x on pairs of subalgebras satisfies

the commutative and associative laws. Two subalgebras B and C of 21 are called

complementary factors if B x C = A9 and a subalgebra β which has at least

one complementary factor is referred to as a factor of 21. For complementary

factors β and C of 21, it follows from the definition of direct product that

b + c = c + b whenever b £ B and c £ C; and aι + (a2 + α3) = (α t + c^) + α 3

whenever two of the elements aί9 a29 a3 belong to one of the pair β and C, while

the remaining element belongs to the other; it also follows that there exist an

{A9 β)-homomorphism / and an {A9 C)-homomorphism g such that a = f{a)

+ g(a) for every a in A.

2This definition of a coset algebra is given in [δ], p. 79 and p. 175. For an intro-duction to equivalence relations and congruence relations, see [3], pp. 159-162.

3This definition of center is given in [5]. It can easily be shown to be equivalentto the definition given in [4], p. 24, when the latter is applied to algebras with one binaryopefation.

ON ALGEBRAS WHOSE FACTOR ALGEBRAS ARE BOOLEAN 2 9 9

The notion of direct product can be extended in an obvious way to any finite

number of subalgebras.

In order to discuss direct decompositions of the algebra 21, that is, the repre-

sentations of A as a direct product of subalgebras, it is important to study

algebraic properties of the family $ (21) of all factors of 21. In particular, it

proves convenient to consider the so-called factor algebra of 21, that is, the

algebraic system formed by the family §(21) and the operation x. (This is clearly

a system in which the closure property does not hold.) An especially simple

case is the one in which the system <^(^)> χ ) ι s what is called a disjunctive

Boolean algebra, that is, in which the operation x has all the properties of

Boolean-algebraic addition (join operation) restricted to couples of disjoint

elements.4 Instead of the term 'disjunctive Boolean algebra', we shall use the

simpler term 'Boolean algebra'. Among consequences which follow from the

assumption that the factor algebra is Boolean, we mention the following: (i)

the factor algebra of 21 has the so-called refinement property; (ii) A has, apart

from order, at most one representation as a direct product of so-called indecom-

posable subalgebras; and (iii) it has just one such representation if it is finite.

Various conditions are known which are necessary and sufficient for the factor

algebra <g (21), x>to be Boolean.5One such condition is that $(2Ϊ) be a Boolean

algebra in the usual sense under set-theoretical inclusion. Another condition,

which will actually be used below, is the distributive law for the factor algebra

in the following form:

// B9 Cx and C2 are factors of 21, and if Cx x C2 exists and is also a factor

of 21, then

B n ( d x C2) = (B n Cx) x (B n C 2),

(where B n C denotes as usual the intersection of the sets B and C).

The main purpose of this paper is to establish two further necessary and

sufficient conditions of the same kind; they will be given in §2, Theorem 3 and

Corollary 4. In particular, from Theorem 3 we shall see that a necessary and

sufficient condition for the factor algebra to be Boolean is that for every factor

B there exists exactly one factor complementary to B. In §3 we shall see how,

by application of the results of §2, various simple and interesting classes of

algebraic systems can be shown to have Boolean factor algebras. Finally, in

§4, an extension of the results obtained to algebraic systems with many oper-

ations will be briefly discussed.

4See [8], p. 205 ff.5See [8], p. 272 ff.

300 J. M. G. FELL AND ALFRED TARSKI

2. Main Results. We start with two auxiliary theorems.

THEOREM 1. Let A = B x C, and let D be a subalgebra of 21. In order that

A = B x D,

it is necessary and sufficient that there be a (C, Bc)~ homomorphism f such

that D consists of all elements of the form c + f (c) with c in C.

Proof. Parti (proof of necessity).6 Assume

(1) A = B x D.

Since by hypothesis

(2) i = ΰ x C ,

there is (see Introduction) an (A, B)-homomorphism g and an (A, C)-homo-

morphism h such that

(3) a = g(a) + h(a) for every a in A .

Obviously h is also a (Z), C)-homomorphism. With the aid of (1) and (2) we can

even show that

(4) h is a (Z), C) - isomorphism.7

Therefore, denoting by k the inverse of the biunique function obtained by re-

stricting the domain of h to /), we can put, for each c in C,

(5) f{c) = g[k(c)].

Evidently

(6) / i s a ( C, 5)-homomorphism.

By (3) and (5),

k(c) = c + /(c) for all c in C.

From this and (4) we deduce that

6 The proof of the "necessity part" of our Theorem 1 is essentially contained in theproof of Theorem 2.17, p. 32 of [4]. Also if, instead of considering arbitrary algebras, werestrict ourselves to the so-called loops, then the "necessity part" of our Theorem 1 caneasily be derived from Corollary 2, p. 69 of [ l ] ; on the other hand, by analyzing the proofof Theorem 1, we can see that the corollary just mentioned extends from loops to arbi-trary algebras.

7See [4], p. 32, the proof of Theorem 2.17.


(7) D consists of all elements of the form c + f(c) with c in C.

Let us fix an element c of C. By (6) and (7),

(8) f(c)C B anάc + f{c) € D.

By (1) there are elements b of B and d of D such that c = d -f b. But d has the

form c t + / ( c i ) , with (^ in C. Consequently, by the associative law for elements

of complementary factors (stated in the Introduction),

c = \-cι + / ( c t ) ] + b = c t + [/(Ci) + i ] ,

from which it follows by the condition of unique representation in the definition

of direct product that c = cx and f{cx) + b = 0, and hence

(9) f ( c ) + b - 0 .

Let bιt b2 G B. By applying again the associative as well as the commutative

law for complementary factors, we obtain from (1), (2), and (8):

U ι + [b2 + f(c)]\ + c = fc, + | [ f c 2 + f ( c ) ] + c\

= bι +{b2 + [c + f(c)]\ = [bt + [ c + f(c)]\ + b2

= ί [ ^ + / ( c ) ] + c{ + 6 2 = [bt + / ( c ) ] + ( c + &,)

= U ί » ! + / ( C ) ] + fc2! + C,

s o t h a t f inal ly

ί * i + i h + f ( c ) ] } + c = { [ ό ι + f ( c ) ] + b 2 l + c .

Hence, by observing that bί +[b2 + f(c)] and [ bt + f{c)] + 62 are in β, while

c C C, we conclude from (2) that

(10) bx + [6 2 + f(c)] = [ ό t + f(c)] + fc2.

Similarly,

(11) 6i + [fc2 + / ( c ) ] = (bt + 62) + / ( C ) .

According to the definition of δ c , (9), (10), and (11) assure us that /(c) C Bc

whenever c £ C , Therefore, by (6),

(12) / is a ( C, Z?c)-homomorphism

By (7) and (12) the proof of necessity is complete.

3 0 2 J M. G. FELL AND ALFRED TARSKI

Part II {proof of sufficiency). Assume that a function /and a set D are given

which satisfy (7) and (12).

Let a £ A. We may write α = c + b9 where b £ B and c G C Since /(c)

G β c , there is a έ t in δ such that f(c) + 6 t = 0. Therefore, applying the as-

sociative properties of elements in the center and in complementary factors, we

get:

a = c + \ [/(c) + bx] + b \ = c + [/(c) + ( bx + 6)] = [ c + /(c)] + ( bx + 6),

where of course c + /(c) G D and bx -\- b C B. Thus every element α of A is of

the form

(13) α = rf'+ ft', where rf'G D and b'£ B.

Suppose that d + & = d ' + ί>', where d, rf' G D9 and 6, i ' G B. By hypothesis,

c? and d' are of the forms respectively c + f(c) and c ' + f(c')f where c, c 7 G C.

Therefore

From this follows c = c', hence d = d'; also

/(c) + ό = / ( c ) + 6'.

Hence, f(c) being in Bc, we obtain, using the commutative law and cancellation

law for elements in the center, b = b\ Therefore:

(14) The representation (13) of an arbitrary a in A is unique.

It is furthermore obvious that

(15) d + b G A for every d in D and b in B.

Finally, suppose d, ί / ' G D , and b9 t ' C δ . As usual, d~c + f{c) and

cf' = c ' + f{c')9 where c, c ' £C. Then, applying various properties of the center

and direct product which were used earlier in this proof, we obtain

(d+ rfO + (& + b') = \Vc + / ( c ) ] + [ c ' + / ( c ' ) ] } + ( 6 + 6 0

= { ( c + c θ + [ f ( c ) + f(c')]\ + (A + A ' )

- ( c + c ' ) + { [ / ( c ) + / ( c ' ) l + ( A + 6 ' ) }

= ( c + c ' ) + i [ / ( c ) + 6 ] + [ / ( c ' ) + A ' ] J

= { c + [ f ( c ) + A ] } + { c ' + [ / ( c θ + A l l

= \[c + / ( c ) ] + A| + { [ c ' + / ( c θ ] + A Ί ,

ON ALGEBRAS WHOSE FACTOR ALGEBRAS ARE BOOLEAN 303

so that finally

( 1 6 ) ( r f + < Γ ) + ( b + V ) = { d + b ) + ( d ' + b ' ) f o r b , V i n B a n d d9 d' i n D .

From (13), (14), (15), and (16) we conclude that

A = B x D9

and the proof of sufficiency is complete,

COROLLARY 2. Let A = B x C. For C £o &e ίλe only complementary factor

of B, it is necessary and sufficient that the only (C, Bc)-homomorphism (with

domain restricted to C) be the constant function f, with f(c) = 0 for every c in C.

Proof. The sufficiency follows immediately from Theorem 1.

To prove necessity, assume that C is the only complement of B, and that g is

a (C, βc)-homomorphism. For any given c in C, by Theorem 1, we have c + g(c)

£ C, and of course also [c + g(c)] + 0 = c + g{c). Hence, by the hypothesis

of the theorem and the definition of direct product, we have g(c) = 0.

THEOREM 3. For the factor algebra of an algebra 21 to be Boolean, it is

necessary and sufficient that every factor of 21 have exactly one complementary

factor. 8

Proof, The necessity follows from well-known properties of Boolean alge-

bras.

To prove sufficiency, assume that every factor of 21 has a unique comple-

mentary factor. Consider three subalgebras β, Cl9 C2 of ?ϊ such that

(1) β, Cι, C2 and Cλ x C2 are factors of 2ί.

Then, for some subalgebras D and C 3 , we have

(2) i = β χ D = ( C 1 x C 2 ) x C 3 ,

and it is known that formula (2) implies

(3) B C {[(Dc xB) n C J x [ ( D c x δ ) n C2]\ x [(Dc x B) n C 3 ] . 9

In view of (2) , there is an {A, Cx )-homomorphism gϊ9 an {A, C2)-homo-

morphism g2, and an {A, C3)-hornomorphism g$ such that

8 This theorem seems to be related to a result in [2], according to which a latticepossessing (relative) complements is a Boolean algebra if and only if all complementsare unique. We see, however, no way of applying the result just mentioned in the proofof Theorem 3.

9See [4], p. 21, Theorem 2.9.


(4) a = [gχ (a) + g2 (a)] + g 3 (α) for all a in 4

(see Introduction). Hence it follows by (3) that

(5) gι(b) CDC x B, g2( b) CDC x B, and g 3(6) € Dc x B for all 6 in B.

Similarly by (2) there is an (A, B)-homomorphism / and an (A9 β)-homo-

morphism f2 such that

(6) « β /jία) + / 2(α) for all a in A .

Clearly / 2g 1 is an (/I, Z))-and hence also a (B, D)-homomorphism. More-

over, if 6 £ B, we have, by (6),

(7) gχ ( b) = / ι g l ( fc) + / ^ ( t ) , where fχ6χ{b) £ B and f2gχ{b)CD.

By comparing (5) and (7), we easily see that f2gι(b) £ Z)% so that f2gχ proves

to be a (B, Dc)-homomorphism. But, by hypothesis, B is the unique factor com-

plementary to D. Therefore, by Corollary 2, f2gx(b) = 0; hence, by (7),

(8) gχ(b) CB for all b in B.

For entirely analogous reasons,

(9) £ 2 ( δ ) £ B for all 6 in B.

Evidently (B n Ct) x (B n C2) exists. By use of (4), (8), and (9) we obtain

B n (Cx x C2) C (B n Q) x (B n C 2 ).

But it is clear that (B n Ct) x (B n C2) C B n (Cγ x C 2 ). Consequently

(10) B n ( d x C2) = (B n Ct) x (B n C 2).

We have now shown that, whenever B, C l 5 C2, and Cv x C2 are factors of

U, the relation (10) holds. As was pointed out in the Introduction, this is a

sufficient condition for the factor algebra of 11 to be Boolean.

COROLLARY 4. For the factor algebra of an algebra 21 to be Boolean, it

is necessary and sufficient that9 for every pair B and C of complementary factors

of 21, the only (C, Bcyhomomorphism (with domain restricted to C) be the con'

stant function f such that f(c) = 0 whenever c £ C.

Proof. This follows from Corollary 2 and Theorem 3.

We should like to conclude this section with some remarks which are related


to Theorem 3, even though they are not connected with the main topic of this

paper. By Theorem 3, in every algebra 21 whose factor algebra is not Boolean,

we can find two different factors B and C which have a common complement D.

Hence it appears natural to consider the relation which holds between any two

factors B and C if and only ΊίBxD = CxD = A for some factor D. This

relation is often involved in the discussion of direct decompositions of algebras

and some of its properties can be found in the literature; it is known, for in-

stance, that any two factors between which this relation holds are isomorphic

(and even what is called central-isomorphic).10 In this connection it may be

interesting to notice that the relation in question, as opposed to that of iso-

morphism, is not transitive, and hence not an equivalence relation. To demon-

strate this fact, we take for a counterexample the additive group 21 = (A, + ) of

complex numbers a + bi, where a and b are integers. If p is an element of A, we

shall denote by [p] the subgroup of 21 generated by the element p. Now let Bχ -

[i]f B2 = [3 + 2i]9 Cγ = [1], C2 = [1 + ι], C3 = [7 + Si]. It is easy to check that

(1) x Cx = Bι x C2 * B2 x C2 = B2 x C 3 .

Let * be the relation which holds between factors B and C if and only if B x D -

C x D = A for some D. Then, by (1),

(2) Cι « C2 and C2 « C3

Now, suppose Ct ** C3. Then there is a factor D of 21 such that A = DxCι = Dx

C3# Using (1), we observe that D « β t ; therefore D = Bu and D is an infinite

cyclic group. Let α, b be such a pair of integers that D = [a + bi]. In order that

D x Cι = A, it is necessary that

a 1

b 0

from which follows b = ± 1. Similarly, in order that D x C3 = A, it is necessary

that

a 7

b 5= ± 1;

1 0 See [4], p. 32, Theorem 2.17, and [ l] , p. 69, Corollary 2.

3 0 6 J. M. G. FELL AND ALFRED TARSKI

this, combined with b = + 1, gives 5α ί 7 = ί 1, an impossibility. Consequently

(3) it is not true that Cx « C3 .

Now (2) and (3) imply at once that « is not transitive.

3. Applications. We shall now apply the results of the preceding section to

various special classes of algebraic systems. Although many of the results we

shall formulate in the present section are not essentially new, it may be interest-

ing to see that all of them can now be established by means of a simple and

uniform method.

While most of our discussion will be based directly on Corollary 4, we shall

start with a simple case of a straight application of Theorem 3.

THEOREM 5. The factor algebra of an arbitrary cyclic group is Boolean.11

Proof. Let ® be a cyclic group of finite order n. As is well known, (3 has

at most one subgroup of any given order. If now B is a factor of ($ of order m,

then a complementary factor of B must have exactly n/m elements, and hence

is uniquely determined. Hence, by Theorem 3, the conclusion follows immediate-

ly. The theorem applies trivially to infinite cyclic groups, since they are known

to be indecomposable.

Theorem 5 can be considerably strengthened if we introduce the notion of a

generalized cyclic group. A generalized cyclic group is a group ® any two

elements of which are members of the cyclic group generated by some third

element of ®. All generalized cyclic groups are Abelian, and finite generalized

cyclic groups are cyclic. Now it is a known result that the lattice of all sub-

groups of a group ® is distributive if and only if © is a generalized cyclic

group. From this it follows that the factor algebra of an arbitrary generalized

cyclic group is Boolean. The latter fact also follows easily from our Corollary

4.

Since the above-mentioned result on lattices of subgroups holds in both

directions, one might enquire whether or not the statement on factor algebras of

generalized cyclic groups admits a converse. One can indeed show easily from

the well-known structure of finite Abelian groups that a finite Abelian group

whose factor algebra is Boolean must be cyclic. But for infinite Abelian groups,

no such characterization of those with Boolean factor algebras is known. In

1 1 For this result, indeed, for the more general result quoted below on lattices ofsubgroups, see [7], p. 267 . The example of an indecomposable Abelian group which isnot a generalized cyclic group, also given below, is closely related to the counter-examples provided by B. Jonsson, On Unique Factorization Problem for TorsionfreeAbelian Groups, Abstract, Bull. Amer. Math. Soc. 51(1945), 364.

fact, there are Abelian groups which are not generalized cyclic groups and which

are indecomposable (their factor algebras being thus trivially Boolean). Such,

for example, is the additive group of all number pairs of the form (3ni + m/2,

5nj + m/2 >, where i, j , m and n are integers.

THEOREM 6. If, in an algebra 2Γ = <^, + >, the only (A, Ac) homomorphism

{with domain restricted to A) is the constant function { with /(α) = 0 for all

a in A, then the factor algebra of 21 is Boolean.12

Proof Let A - B x C, and let k be a (C, Bc)- homomorphism. As we know,

there is an {A, B)-homomorphism g and an (A, C)-homomorphism h such that

a = g(a) + h(a) for all a in A. Then kh is an (A, Bc) homomorphism. Since Bc is

known to be included 1 3 in Ac, kh is also an (A, Ac) homomorphism, and there-

fore, by hypothesis, kh(a)- 0 for all a in A. If in particular a C C, we see

that a = h(a)f and hence &(α) = 0. Now by applying Corollary 4 we obtain the

conclusion.

Using the notion of a centerless algebra, as defined in the Introduction, we

obtain as immediate consequences of Theorem 6:

COROLLARY 7. The factor algebra of a centerless algebra is Boolean.

COROLLARY 8. // 21 = </l, +> is either (i) a centerless group, or (ii) an

algebra in which, for any a, b in A, a + b = 0 implies a = b = 0 (in particular,

if 21 is a lattice1* with zero), then the factor algebra of 21 is Boolean.

Besides centerless algebras, there is another rather comprehensive class of

algebraic systems which satisfy the hypothesis of Theorem 6, which will be

referred to as zero-equivalent algebras. We arrive at these algebras in the

following way: 1 5

Given an algebra 21 = (A, +>, we define recursively the equivalence in order

n between two elements Jα, b in A, this relationship being expressed symboli-

cally by α Ξ b:

1 2 T h e results contained in Theorem 6 and its various applications mentioned below(to centerless algebras, lattices with a zero element, centerless groups, and groupswhich coincide with their commutator subgroups) are stated in [4], pp. 53-55; referencesto earlier related results (for example, of G. Birkhoff and A. Speiser) can also be foundthere.

1 3 S e e [4], p. 26, Theorem 2.11.

4 A lattice is here considered8], p. 200 ff.

1 5 F o r a related construction see [9], § 2 .

1 4 A lattice is here considered to be a system with one operation + (join operation);see [8], p. 200 ff.

(i) a = b if either a-c^-d and b = d + c for some c and o? in A, or else

α = c + (c/+e) and b=(c + d) + e for some c, c/, and e in 4;

(ii) a n=ι b if either c = a and c = b for some c in A, or else α + c = b + c

for some c in .4, or finally a = c -\- e and b = d + e and c = d for some c, α?, and

e in /I.

The elements α and b are called equivalent-in symbols, α = 6-if they are

equivalent in some order n. Remembering now the definition of a commutative

semigroup as an algebra in which the commutative law, the associative law, and

the cancellation law hold, we obtain the following theorem:

THEOREM 9. The relation = is a congruence relation over 2Ϊ; and the coset

algebra 21/= is a commutative semigroup. In fact, = is the smallest congruence

relation R for which 2I//v is a commutative semigroup.

Proof. It is easily shown that the relation = is reflexive, symmetric, and

transitive; and that, if any two of the three formulae a = b, c = d, a + c = b + d

hold, then the third also holds. In particular, the relation = is a congruence

relation over 21, and we can construct the coset algebra 21/=. From the defi-

nition of = and = it follows that a + b = b + a and a + (b + c) = (a + b) + c

for all α, b and c in A. This establishes the commutative and associative laws

for 21/=. Further, from a + b = a + c it follows by the first sentence in this

proof that b = c; this proves the cancellation law. SI/= is therefore a commuta-

tive semigroup.

To prove the last statement of the theorem, it is sufficient to show that if

R is a congruence relation over 21 for which ?!//? is a commutative semigroup,

and a and b are members of A for which a = b, then we have aRb (that is,

< α, by C R). But from a = b follows a = b for some nonnegative integer π. If

n = 0, the conclusion aRb follows immediately from the definition of = and the

properties of ?!//?. The passage from n to n + 1 is carried out in a similar manner.

The algebra 21 is called zero-equivalent if a = b for all a and 6 in Af or

what amounts to the same, if α Ξ 0 for all a in A. The following necessary and

sufficient condition for zero-equivalence is an easy consequence of Theorem 9.

COROLLARY 10. The algebra 21 is zero-equivalent if and only if it cannot

be mapped homomorphically onto any commutative semigroup with more than one

element.

We can point out many different examples of zero-equivalent algebras. For

instance, as is easily seen, an algebra </!,+) is zero-equivalent if to every

element a in A there is an element b such that α + b-b (or i + α= i ) . This

condition will be satisfied, in particular, if in addition to the zero element 0

the algebra has an infinity element oo such that oc + a = oc (or a + oo = oo) for

all a in A. The description of the latter class of algebras assumes a more fa-

miliar form if one uses the multiplicative notation; the algebras to which we

refer are then systems 21 = {A, •) which have a unit element 1 with a 1 =

1 a = α, and a zero element 0 with a 0 = 0 (or 0 a = 0) for every a in A.

Again, an algebra (A9+) satisfies the above condition for zero-equivalence if

every element is idempotent, that is, a + a = a for all a in A. Another interesting

example of a class of zero-equivalent algebras is provided by the groups which

coincide with their commutator groups; in fact, as is seen from Corollary 10, a

group is zero-equivalent if and only if it coincides with its commutator group.

THEOREM 11. The {actor algebra of a zero-equivalent algebra is Boolean.

Proof. Suppose the algebra 21 =* (A,+) is zero-equivalent. Let / be an

{A, Ac)- homomorphism. It was remarked in the Introduction that {Ac, +> is an

Abelian group; consequently any subalgebra of Ac, in particular the subalgebra

onto which / maps A, is a commutative semigroup. Hence, by Corollary 10,

f(a) = 0 for all a in A. The conclusion now follows by application of Theorem 6.

As an immediate consequence of this theorem and the remarks which precede

it, we obtain:

COROLLARY 12. // 21 = (A9+) is either ( i ) an algebra with infinity

element oo such that oo + a = oo (or a + oo = oo) for any a in A, or ( i i ) an algebra

in which a + a = a for all a in A, or ( i i i ) a group identical with its commutator

group, then the factor algebra of 2ϊ is Boolean.

Note that Corollary 12 ( i i ) includes, as does Corollary 8 ( ϋ ) , the case of

lattices with zero element.

To conclude this section, we should like to point out a rather interesting

application of Theorem 1, and specifically of Corollary 2. So far we have been

concerned with algebras 21 = </!, + > having the unique complement property,

in the sense that every factor B of 21 has exactly one complementary factor.

We have seen in Theorem 3 that these algebras coincide with those whose factor

algebras are Boolean. We now turn to algebras which have the unique comple-

ment property in a different sense; in fact, to those algebras 21 = ( / ! , + ) for

which the set A has only one complementary factor with respect to every algebra

33 in which 21 is embedded as a subalgebra and of which A is a factor. By saying

that the algebra 21 == </!,+> is embedded as a subalgebra in the algebra 33 =

<By + ' ) , we mean that A is a subalgebra of S3 in the sense of the Introduction,


and that the operations + and + ' coincide when applied to couples of elements

in A; we assume here that the operation + is not defined for any other couples.

It has been stated that for an algebra ?ί to have the unique complement

property in the new sense, it is necessary and sufficient that ?I be center less . 1 6

The proof of this statement can easily be carried through by means of Theorem 1

and Corollary 2. In fact, the sufficiency of the condition follows almost directly

from Corollary 2. To establish its necessity, we consider an arbitrary algebra

21 = (A, + ) . We can easily embed ϊ as a subalgebra in a new algebra δ =

( By + ' ) in which B = A x C, C being a subalgebra of δ isomorphic to Ac. Let

/ be a function which maps C isomorphically —and hence also homomorphically—

onto Ac. Then D, defined as the set of all c + /(c) , where c £ C, is clearly a

subalgebra of δ ; and we have by Theorem 1 that B = A x D. Now if 21 is not

centerless, Ac contains a nonzero element; therefore /(c) ^ 0 for some c in C.

Hence C £ D, and A has two distinct complements in 33. This completes the

proof. In consequence, the class of algebras having the unique complement

property in the new sense is included in, but by no means identical with, the

class of those algebras whose factor algebras are Boolean.

4. Algebras with many operations.17 Up to now we have considered only

algebras with one binary operation. In this final section we shall state the corre-

sponding results for a large class of algebraic systems with many operations.

A system 21 = {A9 +, 0 0 , Oi9 ••• , O^9 •••> will be referred to as a many-

operational algebra if it is constituted by a set A, a binary operation +, and a

finite or transfinite sequence of operations 0 O , 0l9 , 0g, . The set A

is assumed to be closed under all these operations, and to contain an element

0 which is the zero element for + and is idempotent for each of the remaining

operations Og that is, 0^ (0, 0, , 0) = 0. We restrict ourselves to the case

where all the operations (X are finitary,18 that is, each of them is defined ex-

clusively for finite sequences of definite length n^ (the number n* is called

the rank of O~ , and varies with ξ )•

By a subalgebra of ?I we understand a subset B of A which contains 0 and

is closed under all the operations +, 0 0 , Ol9 , 0ξ9 . If B and C are sub-

algebras of 21, the definition of a (B, C)- homomorphism or (B, C)- isomorphism

is an obvious extension of the earlier case. The definitions of a congruence

relation R over 21 and of the coset algebra 2I/J? are also easily generalized from

the earlier case.

1 6 S e e [4], p. 55. The proof of this statement has not previously been published.

1 7 S e e [4], § § 1 and 2, for most of the definitions given in this section.1 8With small changes, the theory also applies to algebras with infinitary operations.


The definition of center for many-operational algebras is somewhat involved.

Given an algebra 21 = (A, +, Oθ9 0u « , (X, •>, and a subalgebra B of 21, by

the center Bc of B we understand the union of all subalgebras C of 21 which are

included in B and satisfy the following conditions: (i) to every element c in C

there is an element d in C with c + d = 0; (ii) the commutative-associative

formula

(bί + b2) + c = &! + ( b2 + c) = ( ^ + c) + ό2

holds for any b±, b2 in B and any c in C; (iii) the formula

Oξ ( ^ o > K 9 ' y b q _ t , b q + c , b q + l9 ••• , b p Λ m l )

= o ^ ( b 0 , b i 9 . . . , ό p β l ) + σ f ( o , o , . . . , o , c , o , . . . , o )

holds whenever 0*. is an operation of 21 with p — flv, c is in C, io> i> •• > p-i

are elements of B, and q is a nonnegative integer less than p . 1 9 It follows from

this definition that Bc is itself a subalgebra of 21. As before, a subalgebra /Γ of

21, or the algebra 21 itself, is called centerless if Bc = ί 0}, or Ac = { 0 !, re-

spectively.

If 21 is an algebra in the new sense, the notion of the direct product D of two

subalgebras B and C of 21 is defined just as it was in the Introduction for alge-

bras with one operation, except that to conditions (i), (ii), (iii) is added a fourth

condition, namely: (iv) the formula

Oξ (b0 + c 0 , bx + cl9 . . . , bpmml + C p . J

= 0^ (b0, bl9 . . , bp_ί) + 0^ ( c 0 , cl9 . . . , cp^ι)

h o l d s w h e n e v e r CL i s a n o p e r a t i o n of 21 w i t h ρ = n^9 b0, b19 ••• , bp_x a r e e l e -

m e n t s in β , a n d c 0 , c l 5 «•« , Cp^χ a r e e l e m e n t s in C . T h e c o n c e p t s of a factor

of 21, of complementary factors, and of the factor algebra of 21 are defined in

terms of the direct product precisely as they were in the one-operation case.

The concepts of equivalence and zero-equivalence also have a natural ex-

tension to the case of algebras with many operations. We obtain it by modifying

the definition of equivalence of order n (as given in §3 after Corollary 8) as

follows:

(i) a = b if either

The definition of center just given is equivalent to Definition 2.10 of [4], p. 24, ifthe latter is applied to algebras without infinitary operations.

312 J . M. G. FELL AND ALFRED TARSKI

1) one of the two alternatives in the definition of = in §3 holds,

or

2) a = Og (c09 cl9 ... 9 Cpî) + 0~ {dQ9 dl9 ... 9 dpî) and

* = ®ξ ( co + ô > cι + d\ » » cp _ x + dp _ χ )

for some operation (X and some elements c 0 , ••• , Cp_ϊ9 d0, ••• , cL_i in

A (where p = n^ )

(ii) α n = A ί> if either

1) any one of the three alternatives in the definition of = in §3 holds,

or

2) a = 0g {c09 cl9 . 9 cp_x)9 b = Og {dQ9 dl9 ... 9 dp_x)9 and

* / ! « * (i = 0 , l , . . . f P - l )

for some CL and some elements c 0 , ••• , cp-\i dQ, •• , fl?p-i in A (where

p = ^ ) .

As before, a and 6 are called equivalent— in symbols, α = 6— if they are

equivalent of some order n. The algebra .?ί is said, as before, to be zero-equiva-

lent if α = 0 for every a in A.

Corresponding to the notion of a commutative semigroup, which appeared in

the treatment of equivalence for one-operation algebras, here we need the notion

of a commutative semigroup with operators, defined as follows: ?I = (A, +,

OQI 0i9 ••• , 0μ9 > is a commutative semigroup with operators if (i) (.4, +) is

a commutative semigroup, and (ii) the relation

0ξ ( α 0 , al9 ... 9 ap_γ) + 0^ ( 6 0 , bl9 , bp_ι)

= Og ( α 0 + bθ9 ax + bX9 ... 9 ap^γ + bp_ί)

h o l d s f o r a l l 0^ a n d a l l e l e m e n t s α 0 , ••• , « p - i , b 0 , ••• , b p m m l o f A ( w h e r e

p = τiξ ) .

Having extended the basic notions, one can carry through for many-operational

algebras a development similar to that given in §2 and §3 for algebras with one

operation. One then obtains:

THEOREM 13. Theorems and Corollaries 1, 2, 3, 4, 6, 7 and 11 remain valid

for many-operational algebras. Theorem 9 and Corollary 10 hold for many-oper-


ational algebras if in them we replace *commutative semigroup' by 'commutative

semigroup with operators,9

Observe that in the definition of many-operational algebras the first operation

+ plays a special role. On the other hand, it may happen that an algebra 21 =

(A, +, O0 , Oί9 ••• , CL, •••> contains a binary operation Q. which has a zero

element that is idempotent with respect to all remaining operations (+ included).

By permuting, then, the operations +, 0 0 , Oί9 , Ct, ••• so that Ct replaces

+, we obtain a new algebra 21' which is formally different from SI. It turns out,

however, that if the factor algebra of 21 is Boolean, the same holds for 21'. Also

it turns out that if the factor algebra of 21 is Boolean, the same applies to every

algebra 21' obtained from 21 by adding new operations. These two facts are com-

bined in the following theorem:

T H E O R E M 14. Suppose that 21 = (A, +, Oθ9 0l9 •• , Cl, . > and 21' =

{A, + ', OQ, O[y ••• , 0 ' , •••> are two many-operational algebras {having the

same set A), and that the family {+ ', OQ, 0{, , 0*', ! of operations o/ 21'

contains the family {+, 0 0 , 0ί9 ••• , 0^9 . . . i of operations of 21. // the factor

algebra of 21 is Boolean, then that of 21' is also Boolean.20

Proof. The symbols x and x ' will refer to the operation of direct multi-

plication in the algebras 21 and 21', respectively. Let 0 and 0 ' be the zero ele-

ments of + and + ' , respectively.

We define an operation * on the factors of 21'as follows: Let B be any factor

of 21', and let C be some complementary factor of B (in 21'). Let Oβ and 0^ be

such elements of B and C respectively that 0 *= 0B +' 0^. Then B is defined

to be the set of all elements of the form b + ' 0^ with b in B,

Notice that, in case there are several complementary factors of B in 21', the

definition of B* is independent of which one is chosen as C.

We now fix two complementary factors B and C in 21'. We shall show that B

and C are complementary factors in 21. As before, let 0# and 0^ be such ele-

ments of B and C respectively that

0 = 0 β + ' 0 c .

First observe that

(1) θ€ B* n C*.

20This theorem is a simple corollary of some more general results obtained by B.Jo'nsson and A. Tarski but not yet published, concerning algebraic systems which do notnecessarily have a zero element.


Let P be any one of the operations in 21, of rank p; by the hypotheses of the

theorem, P is also an operation in 21'. From the idempotency of 0 with respect

to P and the well-known properties of the direct product (which in future we use

without mention), we deduce that

(2) P(0B, 0 β , . . . , 0 β ) = 0 β

and

(3) P ( 0 c , 0 c , . . . , 0 c ) = 0 c .

Furthermore, if 60, •«• ,6p-i £ B, we obtain from (3) the result that

(4) P{b0 +' 0c, . . . , ^ , _ 1 + ' 0 c ) = P ( έ b . •••, t p - i ) + ' 0 c ;

similarly, if c 0 , ••• , Cp-\ d C, we have

(5) P(c0 + ' 0 β , - . , cp_ί + ' 0 B ) = P(cQ, . . . , c p ^ ) + ' Oβ .

Now (1), (4), and (5) give that

(6) # * and C* are subalgebras of 21.

From the fact that 0 is the zero element of + in A, it is not hard to show

that Oβ and 0^ are the zero elements of + in B and C respectively; that is,

(7) b + Oβ = Oβ + b = b for all b in B

and

(8) c + 0Q = 0c + c ~ c for all c in C.

Using (7) and (8) we find that

(9) (b + ' 0 c ) + (c +' Oβ) = b +' c whenever b € B, c C C,from which it follows that

(10) every element of A can be written uniquely in the form 6* + c*, with 6* inβ* and c* in C*.

Further, if P is any operation of 21, of rank p, and if

&o> ' ' 9 bp^x G B; c 0 , , cp_ι C C,

then by repeated application of (9), (4)> and (5), we get

(11) P\[{b0 + ' 0 c ) + ( c 0 + / 0 β ) ] , . . . , [ ( 6 p _ 1 + ' o c ) + U p - ! + Ό β ) ] |

= P[b0 + Ό C , .-• , fcp_! + Ό C ] + P[c0 + Ό β , ••• , c p _ t + Ό β ] .


Now (6), (10), and (11) give

β* x C* = A.

Thus we have shown that B and C are complementary factors in 21 when-

ever B and C are complementary factors in 21'.

Now let B, Cί9 and C2 be such subalgebras of SI' that

(12) δ x ' C , = B x ' C2 = A.

By our last result, we have

(13) δ * x C* = δ * x C* = A.

But, since the factor algebra of 21 is by hypothesis Boolean, (13) combined with

Theorem 3 (see Theorem 13) gives

(14) C? = C*.

Now using Theorem 1 (see Theorem 13), let g be the ( C l f B c )-homomor-

phism with respect to 21' (Bc being the center of B in SI') such that C 2 is the

set of elements of the form cx +' g(cx) with cx in C\ L e t 0β and 0^ be those

elements of B for which, for some 0^ in Cx and 0£ in C 2 ,

0 = 0 β + Ό C = 0+ + ' 0 + .

Then (14) together with (12) as sures us that

(15) cx + Ό β = cγ +'[g(ct) + ' 0+] for all cx in Q .

In particular, set t ing cγ = 0 ' in (15), we find

0 β = 0+

hence (15) assumes the form

(16) 0 β = g(cx) +' 0β for all cί in Ct .

But by the definition of g9 we have g(cί) C Bc for each cχ in C t consequently

(17) * ! + [b2 +' g(Cι)] = ( 6 t + ό 2 ) + ' [ 0 ' + g ( c t ) ]

whenever o l f 62 C δ and c λ C C t .

Setting Z>2 = 0β and i x = 0 ' i n (17) we conclude with the aid of (16) and (7) that

(18) 0 ' + g(ct) = 0 ' for each cί in Cx .


On the other hand, setting bγ — Oβ and b2 = 0 ' in (17), we get with the aid of

(18) and (7) the result that

g(cι) = 0 ' for all cι in Cγ.

Consequently, by the definition of g, we have

We have now shown that if Bf Ci9 and C2 are subalgebras of 21' for which

(12) holds, then Q = C2 But, by Theorem 3, this implies that the factor algebra

of 21' is Boolean.

We shall end this section with an application of the general theory to rings.

THEOREM 15. The factor algebra of a ring with unit element is Boolean.21

We can prove this theorem by three different methods, applying either Corol-

lary 7, Theorem 11, or Theorem 14.

First proof. Let 21'= (/!,+,•> be a ring with the unit element 1 (and of

course with the zero element 0). Let c £1 Ac By the definition of center, we

must have

(a + c) b ~ a b for all α, b in A.

Setting a = 0, 6 = 1 , we obtain c ~ 0. Thus 21' is centerless, and by applying

Corollary 7 (see Theorem 13) we obtain the desired result.

Second proof. Let 21' = { A9 +, ) be a ring with unit element 1. By the defi-

nition of equivalence of order 0, we have

(a + c) ( b + d) = a b + c d for all α, b, c, d in A.

Setting b = c = 0 and d = 1, we obtain a = 0 for all a in A. 21' is therefore zero-

equivalent, and the desired result follows from Theorem 11 (see Theorem 13).

Third proof. Again, let 21' = {A, +, > be a ring with unit. Let 21 = {A, ).

By Corollary 12 (and a remark following Corollary 10) the factor algebra of 21 is

Boolean. Hence, by Theorem 14, the factor algebra of 21' is Boolean.

It may be noticed that, by applying the first two methods of proof, we can

extend Theorem 15 to wider classes of rings. In fact, the first method of proof

2 1 T h i s result is stated in a different form in [4], p. 55, where a reference to anearlier result of Jacobson can be found. The extension of this result to centerless rings,which will be discussed below, is to be found in [6]; it also follows immediately fromremarks in [4], p. 25 (Example III) and p. 54.

permits us to extend it to all centerless r ings, 2 2 while by the second method of

proof it applies to all zero-equivalent rings. It is easily seen that a ring is

centerless if and only if it has no annihilator different from 0, that is, no element

α / 0 such that a b = 6 a = 0 for all elements b of the ring. On the other hand

a ring proves to be zero-equivalent if and only if every element a of the ring is

of the form

a = b0 c0 + bι ci + ••• + i Λ - i cn-ι ,

where n is a positive integer and bθ9 c09 bί9 ci9 ••• , bn_ί9 cn^ι are elements

of the ring. Either of these classes includes all rings with unit and also many

other rings.

It may be noticed that neither of these two classes of rings includes the

other. Indeed, the ring of even integers is centerless but not zero-equivalent.

To obtain a ring which is zero-equivalent but not centerless, consider the family

R of all finite sets of real numbers x for which 0 < x <_ 1. If A9 B ζl R9 we

denote by A + B the symmetric difference of A and B9 that is, the set of numbers

belonging either to A or to B but not to both; and we denote by A B the set

of all numbers x for which (i) 0 < x £ 1 and (ii) the number of elements y of A

such that x — y €1 B is odd. The system 5? = </?,+, ) can be shown to be a

ring which is zero-equivalent but not centerless. Indeed, 5R has the further proper-

ties that it is commutative and every element in it is of order 2.

Examples of rings whose factor algebras are not Boolean are easy to obtain.

Let 21 = (A9 +> be an Abelian group with zero element 0, and set a b = 0 for

all α, b in A. Then 21' = {A9 +, > is a ring whose factors are identical with

those of ?I. Thus, if the factor algebra of ?I is not Boolean (for example, if ?I is

the four group, that is, the direct product of two groups of order 2), then the

factor algebra of 21' also is not Boolean.

R E F E R E N C E S

1. R. Baer, Direct decompositions, Trans. Amer. Math. Soc. 62(1947), 62-98.

2. G. Bergman, Zur Axiomatik der Geometrie, Monatsh. Math. Phys. 36(1929), 269-284.

3. G. Birkhoff and S. MacLane, A survey of modern algebra, Macmillan, 1941.

2 2 I t should be noticed that our definition of the center of a ring, obtained by special-izing to rings the general definition of the center of an algebra with many operations,differs from the usual meaning of that term.

318 J M. G. FELL AND ALFRED TARSKI

4. B. Jonsson and A. Tarski, Direct decompositions of finite algebraic systems,Notre Dame Mathematical Lectures, No. 5, 1947.

5. , On direct products of algebras. Abstract 149, Bull. Amer. Math. Soc.51(1945), 656.

6. A. Kurosh, ίzomorfizmy pryamyh razlozenii {Isomorphisms of direct decompo-sitions), (Russian; English Summary), Izvestiya Akad. Nauk S.S.S.R. Ser. Mat. 7 (1943),185-202.

7. O. Ore, Structures and group theory, II, Duke Math. J., 4(1938), 247-269.

8. A. Tarski, Cardinal algebras, with an appendix by B. Jonsson and A. Tarski,Cardinal products of isomorphism types, New York, 1949.

9. A. Tarski, Algebraische Fassung des Massproblems, Fund. Math., 31(1938), 47-66.

UNIVERSITY OF CALIFORNIA, BERKELEY

SYMMETRIC PERPENDICULARITY IN HILBERT GEOMETRIES

P. J. KELLY AND L. J. PAIGE

1. Introduction. A hilbert plane geometry [2] can be generated in the follow-

ing way. Let K be a simple, closed, convex curve in the euclidean plane and //

its open interior. If a and b are any two points in H, they determine a line a x b1

which intersects K in a pair of points u and v. With R denoting cross-ratio, the

hilbert distance from a to b is defined by

h ( a , b ) = k I l o g R ( a , b; u, v ) \ 9

where k is an arbitrary positive constant. The region H is then a metric set with

respect to K. Under the additional requirement that K contain at most one seg-

ment, H defines a hilbert plane geometry in which any pair of points are uniquely

connected by a geodesic, and these geodesies are open straight lines. If K is

an ellipse, then the hilbert geometry coincides with the well-known Klein model

of hyperbolic geometry.

Perpendicularity in H is defined through the idea of distance. If p and ξ are

any point and line respectively, then a point f on ζ is a "foot of p on ζ" if

h(p, f) <h(p,x) for all points % on ξ. A line η, intersecting ξ, is perpendicular

to ζ if every point on η has the point of intersection, ζ x η,* as a foot on ζ.

Under this definition, there is no need for the perpendicularity of η to ξ to

imply the perpendicularity of ζ to η. The aim here is to show that when perpen-

dicularity is always symmetric, the hilbert geometry is hyperbolic.

As before, let p and ξ be any point and line in //, and let η be a line passing

through p and intersecting K in the points u and v. It can be shown quite simply

that a necessary and sufficient condition for η to be perpendicular to ξ is that

a pair of supporting lines exist, one at u and one at v, intersecting at a point w

on ξ [ l ] . If η is perpendicular to ξ, then the previous statement implies that η

is also perpendicular to every line through w which is a secant to K. When such

a secant cuts K at points m and n9 then symmetry of perpendicularity requires

that a supporting line exist at m, and one at /ι, such that the two intersect on η.

1Here and henceforth the line joining a and b will be indicated by a X b, and sym-metrically the point of intersection on lines ξ and η by ζ X η.

Received February 1, 1952.Pacific J. Math. 2 (1952), 319-322

319

320 P. J. KELLY AND L. J. PAIGE

2. The family F. The foregoing facts suggested the independent problem of

identifying the following family of curves.

Family F: Every curve C in F is a simple, closed, convex curve. If sup-

porting lines at p and q on C meet at w, and ξ is any secant through w,

cutting C at m and n9 then supporting lines at m and n exist, intersecting

on p x q.

We are going to show that the family F consists of all triangles and ellipses.

LEMMA 1. If a curve C in F contains a straight line segment then the curve

is a triangle.

Proof. Let a and b denote the end points of a segment contained in C, and

take p to be a regular point (a point of C with unique supporting line) of C which

is not on ζ — ax b. If σ denotes the supporting line at p, assume that q = σ x ζ

is not a point of C. Because the secants to C through q form a continuum, while

the corner points (points possessing more than one supporting line) of C are

denumerable, there exists a secant 77, through q, such that its intersections with

C are two regular points m and n. But by the definition of C, the unique sup-

porting lines at m and n must intersect on p x b and also on p x α. Hence they

intersect at p, which contradicts the fact that p is regular. Therefore the inter-

section σ x ξ is a point of C, and so is either a or 6. Suppose it to be α. The

segment from α t o p is then part of C. Let c denote the other end of the largest

segment contained in C and containing the segment from α t o p . Let r be a regu-

lar point of C, not on ξ nor on γ = a x p, and let δ be the supporting line at r.

By the same reasoning as before, the points ^ x 5 and 8 x γ must lie on C, and

hence are the points b and c respectively. Thus C is the triangle α, b9 c.

LEMMA 2. // a curve C in F contains a corner point, then the curve is a

triangle.

Proof. Let p be a corner point on C, with δ t and δ 2 two supporting lines at

p. Assume: (*) that no supporting line contains two points of C. Let q and r be

any two regular points of C, with σ and 77 denoting their respective supporting

lines. Set iίj = σ x δj ( i = 1, 2) and v = η x (p x q). Because of (*), the

line ui x r is a secant, and intersects C again at a point s{ (i = 1, 2) . By the

definition of C, at s, a supporting line exists which intersects 77 at a point of

p x q9 namely at the point v. But because of (*), the point v is exterior to C.

In addition to 7/, then, there can be only one other supporting line through v*

Hence the lines v x sx and v x s2 are the same, which contradicts (*). Be-

cause (*) is false, C contains a segment, and so, by Lemma 1, it is a triangle.

SYMMETRIC PERPENDICULARITY IN HILBERT GEOMETRIES 321

THEOREM 1. The family F consists of all ellipses and all nondegenerate

triangles.

Proof. If C contains a segment or a corner point then it is a triangle; so

suppose C to be strictly convex and differentiate. Let pί and p 2 be two points

of C such that the supporting lines, σι and σ 2 , at p t and p 2 are parallel. Intro-

duce a rectangular reference frame so that pi is the origin, σ t is the y-axis, and

with p 2 lying in the first quadrant. Take θ to denote the acute angle between the

the line η = p 1 x p 2 and the x-axis, and let σ2 be the line x = k. A vertical

chord of C, lying on the line σ(x) through (x, 0), is cut by η into an upper and

lower segment such that the ratio of their lengths, μ(x), is constant for all x

on the interval <0, k). To prove this, let T be the affinity y ' = — x tan θ + y,

χ/ = x, taking C into a new convex curve C'. Under Γ, the line η goes into the

%-axis, which seperates C" into an upper curve yι - fι(x) and a lower curve

Ϊ2 = f2^x^' Because T preserves distance on any vertical line, the ratio μ(x)

equals fχ {x)/f2 {%)> The line σ{x) is a secant to C through σι x σ2 hence the

tangents to C, at its intersections with σ{x)9 are lines which intersect on η .

Then C" has the property that the tangents at {x, fγ (x)) and (x, f2 {x)) inter-

sect on the x-axis. From simple triangle relations it follows that

for Λ; on < 0, A: >. If a is fixed, and x is variable, on < 0, k ) , then the equality

Ja fix) Ja fix)

shows that

and hence that μ(x) is a constant. The original curve C, therefore, has the

property that if a line -joins the contact points of two tangents which are parallel

in a direction Ot, then the line cuts all chords parallel to α in a ratio which is

constant (with α). But then it is known that the ratio is unity for all directions

and that the curve is an ellipse [3]. Since it is easily shown that either a tri-

angle or an ellipse does belong to the family F, the theorem is complete. In

particular, it may be noted that the property of family F, applied to strictly

convex curves, characterizes the ellipse.

3 2 2 P. J. KELLY AND L. J. PAIGE

3. Symmetric perpendicularity. The answer to the original problem is now

clear. When perpendicularity is symmetric in a hilbert geometry, then the curve

C belongs to the family F. Since C can have at most one segment, it is not a

triangle, and hence is is an ellipse. Therefore the geometry is hyperbolic. Thus

we have proved the following theorem.

THEOREM 2. The hilbert geometries in which perpendicularity is symmetric

are the hyperbolic geometries.

The result obviously extends to higher dimensions. Perpendicularity refers

to lines in the same plane. When the perpendicularity is symmetric, every plane

section of the gauge surface K is an ellipse; hence K is an ellipsoid which

defines a higher dimensional hyperbolic geometry.

REFERENCES

1. H. Buseman and P. Kelly, Projective geometry and projectiυe metrics (In Press),Chapter 3, Section 7.

2. David Hilbert, Grundlagen der Geometrie, Anhang I, Seventh Edition, Leipzig,1930.

3. Tadahiko Kubota, On a characteristic property of the ellipse, Tohoku Math. J. 9(1916), 148-151.

UNIVERSITY OF CALIFORNIA, SANTA BARBARA

UNIVERSITY OF CALIFORNIA, LOS ANGELES

ON A CHARACTERISTIC PROPERTY OF FINITE SETS

G. KϋREPA

1. Introduction. There are several equivalent definitions of finite sets

[2], [5] The purpose of this note is to give an equivalent property of finite sets

in terms of ramifications of sets.

DEFINITION 1. A partially ordered set S= (S; £ ) is said to be ramified

or to satisfy the ramification condition ([3], pp.69, 127; cf. 4) provided that

for every x ζl S the set (— 00, x) of all y C S satisfying y < x is totally ordered

(that is, contains no distinct noncomparable points). If the points of a ramified

set (S; <) are the same as these of a set M, one says that (S; < ) is a ramifi-

cation of M.

DEFINITION 2. A chain (anti-chain) of a partially ordered set (S; <_) is

any subset of S containing no distinct incomparable (comparable) points. Every

set containing a single point is considered both as chain and as anti-chain.

DEFINITION 3. For a partially ordered set (S; < ) = S, we denote by

(1) 0(S) or OS

the system of all maximal chains contained in S; analogously,

(2) 0 ( S ) or OS

denotes the system of all maximal anti-chains of 5.

THEOREM. In order that a nonvoid set S be finite, it is necessary and suf-

ficient that for every ramification T(S) of S the relations

(3) MCOT(S), ACOT(S)

imply

(4) M n A ^ A ( Λ = vacuous s e t ) .


323

324 G. KUREPA

2. The condition is necessary. Otherwise, there would be a finite set S, a

ramification T(S), a set M £ OT(S), and a set A £ 7)T(S), such that

(5) H M = Λ,

Now, A is a maximal anti-chain of T(S); consequently, for every x £ T(S)

there is a point a(x) £ A such that the set i x, a(x)} is a chain of T(S). (Other-

wise, the set 4 u U ! would be an anti-chain greater than the maximal anti-

chain A,)

In particular, for any x £ M £ 0 T(S), the points x, a(x) are comparable.

We say that

(6) x < a(x).

Since M is a maximal chain of the ramified set T(S), M is an initial portion of

T(S); that is, M, which contains the point x contains also every point of T (S)

preceding x. In particular, if (6) did not hold then M would contain also a(x) <

x; consequently, a{x) £ M n A, contrary to the assumption (5).

Thus if (5) held then for every x £ M one would have (6); but M, as a non-

void subset of the finite set Γ(S), would have a terminal point, say I; I would

be a final point of T(S), too, contrary to the relation (6) for x - I. Thus the

relation (5) is not possible.

3. The condition is sufficient. If for every ramification T(S) the relations

(3) imply (4), then the set S is finite. Otherwise, the set S would be infinite;

consequently, there would be a one-to-one correspondence φ of the set N of

all natural numbers into S. Now, let us define the ordering (S; <_) by trans-

plantation of a certain order of the set N. We shall order N according to the

scheme \

1—>3 —»5 —»7—> .Nι ^ N N»

2 4 6 8 . . . .

That is, the set 2iV— 1 of all 2n — 1 (n £ N) is ordered as in the natural

order; for every n £ /V, the set of numbers preceding 2n consists of the numbers

2v—l {v = 1, 2, , n); all other couples of natural integers are incompar-

able, by definition. In the ramified set /Vo so obtained one sees that 2/V £ ON0,

that 2ZV-1 £ ON0, and that the sets 2/V, 2W - 1 are disjoint. Now, the set

S being infinite by hypothesis, there is a one-to-one mapping φ of N = No into S.

1For the definition of schemes or diagrams of partly ordered sets see Birkhoff [1,p .6] .

ON A CHARACTERISTIC PROPERTY OF FINITE SETS 325

That enables us to define the order in S by transporting the order of NQ into

S so that, on the one hand, the mapping φ is a similitude between No and φNQ C_

S, and so that, on the other hand, no point of φNo is comparable to any point

of 5\0iV o, and so that S\<£/Vo contains no comparable couple of distinct

points.

It is obvious that the set (S; <_) is ramified, that the set φ{2N — 1) is a

maximal chain of (S; <)9 and that the set A - φ(2N) u (S^φN) is a maximal

anti-chain of (S; < ).

According to (4), the set A n φ(2N — 1) would be nonvacuous, contrary to

the fact that the sets A, φ(2N - 1) are disjoint.

Thus, the proof of the theorem is completed.

4. Observation. We observe that the condition of ramification in the state-

ment of the theorem is essential. Namely, if we consider the partially ordered set

Sx = {1, 2, 3, 4, 5} with the diagram

3

ΐ1 2

it is obvious that { 2, 3, 5 } is a maximal chain of S, that { 1, 4} is a maximal

anti-chain of S, and that the set { 2, 3, 5 } does not intersect the set { 1, 4 }.

5. Questions. In connection with the statement of the theorem it is interest-

ing to consider the following two questions:

QUESTION 1. Is there a partially ordered nonvacuous set S such that there

is no maximal anti-chain A £ 0 5 satisfying A n M ^ Λ for every maximal chain

M COS?

QUESTION 2. Is there a partially ordered nonvacuous set S such that there

is no maximal chain M £ 0 S satisfying M n A ^ Λ for every maximal anti-chain

AC OS?

REFERENCES

1. G. Birkhoff, Lattice theory, New York, 1948.

2. K. Iseki, Sur ίes ensembles finis, C. R. Acad. Sci. Paris 231 (1951), 1396-1397.

3. G. Kurepa, Ensembles ordonnes et ramifies, These, Paris, 1935; Publ. Math. Univ.Belgrade 4(1935), 1-138.

4. , L'hypothese de ramification, C. R. Acad. Sci. Paris 202(1936), 185-187.

326 G. KUREPA

5. A. Tarski, Sur les ensembles finis, Fund. Math. 6(1924), 45-95.

MATHEMATICS INSTITUTE OF THE UNIVERSITY

ZAGREB, YUGOSLAVIA

A DIOPHANTINE PROPERTY OF THE FUCHSIAN GROUPS

JOSEPH LEHNER

1. Introduction. A central result in diophantine analysis is the following [4]:

For every irrational real number OC and every positive constant h < y 5/2

there exists an infinite sequence of reduced fractions pi/qi such that

(1) α - < (i = 1, 2, 3, . . . ) •

The property obviously fails if α is rational.

This result can be related to the theory of the (elliptic) modular group [3] We

sketch the proof for a smaller value of h. The fundamental regions of the modular

group lie in the upper half-plane and have cusps or vertices at every real rational

point. Thus the fractions pi/qi are the parabolic vertices of the modular group,

while α is a limit point of vertices which is not itself a vertex. To establish

(1), take a vertical line L in the upper half-plane ending at OC. Then L cuts

infinitely many fundamental regions. Select a point *£ = &£+ 0 ^ lying on L in

the region /?, (k - 1, 2, 3, ); there is a unique modular substitution

%z ~ Pk

which carries R^ and therefore z^ into the standard fundamental region Ro:

(\z\ > 1, | R z | < 1/2).

We have z£ = zkVk = %k + iyk and yζ = Ύk/Q> where

Now y£ > y/~3/29 since this is true for every point in Ro; equality is attained

only for points with rational abscissae. Since the arithmetic mean is not less

than the geometric, we have

k > Q>2\qkyk\ | ^ α - p J ,

Received December 3, 1951.

Pacific /. Math. 2 (1952), 327-333

327

328 JOSEPH LEHNER

which is (1) with h = y/S/2. This result was first obtained by Hermite [2]; he

used algebraic methods.

The above proof depends only on the circumstance that there is a funda-

mental region of the modular group which is at a positive distance from the real

axis. This suggests a generalization to Fuchsian groups whose fundamental

regions have the same property. But the result can be proved for more general

groups. We are not particularly interested in the best possible value of A, but

only in the existence of a constant h depending only on the group and not on

α.First, let us note that in a Fuchsian group with a finite number of generators

and with real axis as principal circle, the set of parabolic points (which we

shall assume includes the point oo) consists of real numbers, of the form p/q,

p real, q > 0. Moreover, p and q have distinguishable identities; that is, they

cannot be replaced by λp, λq. For the number of inequivalent parabolic vertices

of the group is finite; p/q must be equivalent Jto one of them, say oo. Then the

group contains a substitution carrying p/q to oo; that is, the first but in general

not the second of the substitutions

belongs to the group. Thus the statement that (1) is satisfied by infinitely many

of the parabolic points p^/q of a Fuchsian group has a nontrivial content.

Our result is now as follows:

THEOREM. Let G be a Fuchsian group, with real axis as principal circle,

which is generated by a finite number of substitutions. Let P, the set of para-

bolic points of Gf be an infinite set including the point oo. Let (X £ P -P.

Then there exist pjq^ (ι = 1, 2, •••) for which (1) is true9 where p^/qi £ P>

and h = h{G) is a geometrical constant depending only on the group G.

That oo is a parabolic point of G is no real restriction since the group can

always be transformed by a linear transformation (G—>T~ιGT) to secure this.

We intend to develop further arithmetic properties of the Fuchsian groups in

future investigations. A generalization of the semi-regular continued fractions

is indicated.

2. Preliminaries. In this section we gather together some known facts con-

cerning the fundamental regions of Fuchsian groups which will be needed in §3.

Reference may be made to Ford [ l ] .

We shall find it convenient to transform G into Γ = S ιGS, choosing S so

that no substitution V of Γ has infinity as a fixed point, and so that the real

axis and upper half-plane are mapped into the unit circle (Q) plus its interior.

A DIOPHANTΪNE PROPERTY OF THE FUCHSIAN GROUPS 329

Then every substitution

γz + 8

has γ -=f 0 and possesses the isometric circle \γz + 8 \ = 1, which is orthogonal

to Q. Furthermore, in any infinite sequence of isometric circles, the radii

(= γ~ι) tend to zero.

The fundamental region Ro of Γ is that part of the interior of Q which is

exterior to all isometric circles; it is a simply connected region, bounded by

arcs of isometric circles, which are called sides. Two sides intersect if at all

in a single point (vertex) which may lie in the interior of Q or on Q; in the latter

case the sides must be tangent. The boundary of Ro on Q consists entirely of

vertices if Γ is a Fuchsian group of the first kind; in a group of the second kind,

it contains arcs of Q; it may also contain vertices. These arcs are not sides of

Ro; they will be called bounding arcs. A side which, together with its endpoints,

lies entirely inside Q is called an open side.

The transformations Vi of Γ map Ro into fundamental regions /?j, which in

their entirety, fill the interior of Q completely and without overlapping, and

cluster in infinite number about each parabolic vertex.

The sides of Ro are arranged in conjugate pairs (Zy, Zy'); that is, there is a

substitution Tj of Γ carrying lj into Zy. (If a side is followed immediately by its

conjugate, the two sides are part of the same isometric circle.) Starting with

any particular vertex in RQ9 we label the sides and bounding arcs 1, 2, 3, ,

in counterclockwise order; as the conjugate of a side appears, it is labelled with

the corresponding primed integer. (The bounding arcs are not considered to have

conjugates.) As an example, the usual fundamental region for the modular group

would be labelled 1, 2, 2', l ' , if we start from the vertex corresponding to

infinity in the upper half-plane.

Since each region /?; is a directly conformal image of Ro, the sides are

described in the same order in both. The side of /?/ which is the image of the

side k in Ro is written Ic. When two regions abut, the common side bears two

labels, one for each region.

Suppose two regions /?;, Rj abut along a side V in /?;. The substitution

carrying Rι into Rj carries some side, evidently Z, into Z'. Therefore, the side of

Rj common to /?,-, being the image of Z in /?;, is labelled Z in Rj and, as we have

seen, is labelled Γ in /?;. When two regions abut, then, the common side bears

labels in the two regions which are conjugate to each other.

We are now able to define a quantity η = 77 ( Γ ) which in §4 will be related

to h(G) of (1). If Z is any side of Ro> let dι be the maximum of the distances

3 3 0 JOSEPH LEHNER

of points of I from Q. If I is a bounding arc, construct the unique circle orthogo-

nal to Q which passes through the endpoints of I; then dj is the largest distance

from Q of points of the arc of this circle lying in Q. Now we define

(2) η(Γ) = min d{ ,

where I runs over the (finite number) of sides and bounding arcs of Ro. Obviously

(3) τ / ( Γ ) > 0 .

3. Lemma. Denote by Pι the set of parabolic vertices of Γ, and let β C

Pί — Pί Plf and therefore β, lies on Q. Denote by π the image of oo on Q under

the transformation S (§2). Let λ be the unique orthogonal arc connecting π and

β. Since β is not in Pί9 λ crosses infinitely many regions, say Ri9 R29 ••• > in

that order. Let λj be the closed arc of λ lying in R(. The substitution Fj ι maps

Ri on Ro and λj on an arc λj' = λiVf1 lying in Ro. Our object is to prove the

following:

LEMMA. For each β there exist infinitely many values of y, and correspond-

ing points tj on λy whose images t/ on λy are at a distance from Q greater that

η ( Γ ) . Further, £y—>/3.

To this end let us study the sequence of sides of the fundamental regions

crossed in turn as λ is traversed from π to β We may just as well consider the

maps of these sides in Ro and thus write the sequence as

Λ p K\9 & 2 ? n>2* ™ 3 ' * * * *

(It is understood that all letters appearing stand for either unprimed or primed

integers, and (/&')'= n.) Since the first side in each pair is merely the conju-

gate of the last side in the preceding pair, we may abbreviate the above to

(4) ki9 k2, k 3 >

where only the last side crossed in each region is written down. This sequence,

which is uniquely determined by β, will be denoted by (β ).

It is clear that (β) must contain some pair (k> I) infinitely often, for Ro has

only a finite number of sides and arcs. The appearance of k in the (y — l ) t h

place of (β ) followed by I means that, in /?y, λy starts from the side A;'and ends

in the side Z. Then λy starts from the side k' of Ro and ends in the side I of Ro

and is orthogonal to Q.

Suppose first that &' and I are not consecutive sides of Ro in the labelling

already adopted, so that there is at least one side or bounding arc m following

A DIOPHANTINE PROPERTY OF THE FUCHSIAN GROUPS 3 3 1

k' and preceding /. Then λy, being orthogonal to Q and cutting only k' and Z,

must lie between m and the center of Q. There is a point tj on λj whose distance

from Q exceeds dm; and hence exceeds 77 ( Γ ) . This is obvious if m is not a

bounding arc. Otherwise, the circle which determines dm lies completely interior

to λy'. Hence, in all cases, the image of tj is a point tj in λy with the desired

properties. Since / assumes infinitely many values, the lemma is proved under

the above mentioned supposition.

According to this result, the lemma may be established by demonstrating the

existence of a pair (x, y) occurring indefinitely often in the sequence (β) such

that x'9 y are not consecutive sides of Ro. This we proceed to do.

We divide the discussion into two cases:

A. Ro has all vertices on Q. Let (k, x) be an infinitely recurring pair in

(β). If there is an x such that k'9 x are not consecutive, apply the previous

result. Otherwise, from a certain point on in (/3), k occurs followed always by

/ where k'9 I are consecutive. Now consider infinitely recurring triples {k9 Z,Λ).

If an x exists for which Z', x are not consecutive, there is nothing further to

prove. If not, all late triples in (β) involving k, I are (k9 Z, m), where V9 m are

consecutive.

In the continuation of this process, two possible situations can arise. We

may find a block (A;, Z, , r, s ) occurring infinitely often in {β ) with r', 5 not

consecutive. If this does not happen, then we arrive at an infinitely recurring

block (k9 Z, ••• , u9 v) with k'9 I consecutive, ••• , u'9 v consecutive, and υ\ k

consecutive. This block may contain all or only some of the vertices of Ro. But

now these blocks must occur in succession with no entries in between in (j8).

For if we assume z £ k follows v infinitely often in (β)9 we have the recurring

pair (v, z), and (v'9 z) are not consecutive since v'9 k are.

The sequence (/3) now looks like this:

a l 9 α 2 , ••• , a^-^ B> B> B 9 ••• ,

where B is the block (k9 Z, , u9 v). The sides k\ I of the first block B lie in

RN + ί Let the vertex at the intersection of these sides be K. Then K lies on Q9

and k' and I issue from K. The next side cut by λ is ~m. Since Z'and m are con-

secutive in R09 it follows that V and m issue from K. Continuing in this way, we

see that from a certain point on all the sides cut by λ issue from K. Let k be

one of these sides; k is an arc of an isometric circle which cuts Q in two points,

K and another point Kγ\ and β lies in the interior of the arc KKi In fact, β lies

inside all such arcs formed from the sides of all regions cut by λ. Since the radii

of the isometric circles tend to zero, the distance from β to K is arbitrarily

332 JOSEPH LEHNER

small. This contradicts the hypothesis that β is not in Px.

B. Ro has some vertices not on Q. Let k', I be sides of Ro meeting at a

vertex K which lies on Q. Suppose {k\ I) recurs indefinitely in (/3). We con-

tinue the argument as in case A. The side Z is not open since it meets ά ' a t / L

Hence, V is not open. The side m consecutive to / ' i s thus not open. Therefore,

the block (k9 Z, m, ••• , u, v) consists entirely of sides which are not open.

Then we are in case A.

If (k\ I) does not occur infinitely often for any nonopen k\ then all pairs

(x, y) which do recur indefinitely consist of open sides. Then the argument is

similar to that of case A except that the vertex K lies in the interior of Q. Infin-

itely many sides cut by λ issue from K. This is known to be an impossibility:

since K i s a vertex in the interior of Q, only a finite number of sides of funda-

mental regions can end at K. Hence the possibility envisaged cannot actually

occur.

This completes the proof of the lemma.

4. Proof of the theorem. Let Q* be an interior circle concentric with Q and

at a distance η(Γ) from Q. The content of the lemma is that on the arc con-

necting the fixed vertex π to any point β C P t - P x there exists a sequence of

points ti9 t29 —> β whose transforms t{> ί2> iR ^o ^ e inside Q*

Recalling that Γ = S~ιGS9 we map Q back onto the real axis E by means of

S. Then the interior of Q is mapped onto the upper half-plane H and π is carried

to infinity, Pt to P, and β C Pι - Px to α € P - P. The sides of Ro in Q

are mapped into sides of Ro' in //, some of which may extend to infinity. The

bounding arcs become intervals of E of finite length. The image of λ is a verti-

cal line L ending at GC, and Q* is carried into a circle E* whose lowest point is

at a distance h(G) above E; we use this to define h(G).

The sequence tn —> β is mapped into a sequence zn —> (X, zn on L. The

lemma then states that the sequence zn has transforms zn' in Ro which lie inside

£**. The distance of zπ'from E is therefore greater than h; that is,

(5) h ; > h u = l, 2 , 3 , . . . ) .

Let \qn ~qn) be the substitution of G carrying Rn into Ro; then pn/qn is

carried to infinity, and pn/qn is thus a parabolic vertex.

Now applying the argument in §1, we see that pi/qi (i = 1, 2, 3, ,)

satisfies (1) with h = h{G) .

A geometrical definition of h can be given. For each side I of /?0, set

d<ι (< + oo) equal to the maximum height of points on Z. If Z is a bounding arc, set

dι equal to half the length of Z. Then

A DIOPHANTINE PROPERTY OF THE FUCHSIAN GROUPS 333

(6) h(G) = min dt > 0 ,i

where I varies over the sides and bounding arcs of Ro'.

REFERENCES

1. L. R. Ford, Λutomorphic Functions, McGraw-Hill, New York, 1929.

2. C. Hermite, Sur Vintroduction des variables continues dans la theorie des nombres,J. Reine Angew. Math. 41(1851), 191-216.

3. G. Humbert, Sur la methode d*approximation d'Hermite, J. Math. Pures Appl. (7) 2(1916), 79-103.

4. A. Hurwitz, Uber die Angenaherte Darstellung der Irrational zahlen durch rationaleBrϋche, Math. Ann. 39(1891), 279-285.

UNIVERSITY OF PENNSYLVANIA

GROUPS OF ORTHGONAL ROW-LATIN SQUARES

D O N A L D A. N O R T O N

1. Introduction. An n by n square array of n2 elements, consisting of n dis-

tinct elements each repeated n times, will be called a pseudo-latin square. If

each row contains each distinct element once, the square will be called row-

latin. Correspondingly a column-latin square will be one where each column con-

tains each distinct element once. A square which is both column-latin and row-

latin is a latin square. From two pseudo-latin squares A and B, a composite

square may be constructed by superimposing the square B on the square A. If

the composite square contains each of the n2 possible distinct pairs, the square

A is orthogonal to the square B9 and the resulting square is called greco-latin.

In this paper a product operation for row-latin squares is defined analogous

to that of Mann [ l , p. 418] for latin squares. It is shown that under this opera-

tion the set of row-latin squares forms a group. It is further shown that the

existence of sets of mutually orthogonal latin squares depends on the parallel

problem for row-latin squares so that existence problems of latin squares may

be studied in the light of row-latin squares.

In §§3 and 4 some of the sets of orthogonal row-latin squares which arise

from this product operation are studied.

2. Row-latin squares. To each theorem concerning row-latin squares, there

is an immediate dual theorem concerning column-latin squares; this will not be

given, but the reader can easily supply it.

Let the distinct elements of an n by n row-latin square be designated by the

natural numbers 1, 2, , n. Then the i th row of the square determines a permu-

tation 5β/ of these numbers from their natural ordering. The square is completely

determined by giving the permutations (^ϊ9 •••, 5βn ) defined by the rows 1,

2, , n respectively. The product of two row-latin squares A and β, which

describe permutations (5β t, , tyn ) and ( Ώί, , Ώn ), may be defined as the

square C - AB = (5pt Q 1, , ^n ϋn ) whose i th row is given by the product

permutation ^ Ωj . The product of two permutations of n elements is a permu-

tation of the same elements, so the product of two row-latin squares is a row-

Received January 23, 1952. The author wishes to thank the referee for suggestingthe use of Lemma 2 to simplify several proofs.

Pacific J. Math. 2 (1952), 335-341

335

336 DONALD A. NORTON

latin square.

THEOREM 1. The set of all row-latin squares is a group of order (n\)n.

Proof. Let 3 be the identity permutation,

s = ( 1 2 ").\l 2 nl

Then the square1 2 . . . n1 2 n

I = ( 3 , 3, . . . , 3) =

•1 2 n

is the unit element of the group. If 5β ι is the inverse permutation of 5β , then the

square

is the inverse square of 4 in the group. The group is not in general commutative

since the permutation group is not commutative. Each row of a row-latin square

may be constructed in nl ways; and since the n rows are independent of each

other, the order of the group is (nl)n.

COROLLARY la. The group of all row-latin squares is isomorphic to the

direct product group of n permutation groups, each on n elements.

To prove the corollary it is sufficient to note that every row-latin square

A = (5$!, ••• 9 tyn) can be written as the product of squares A{ ' A2

# # * An,

where Aι = ( 3 , 3, , 3, $βj, 3, , 3 ) with the permutation 5βt in the i th posi-

tion. Moreover, the set of all squares (3 , , 3, 5β, 3, , 3 ) , where 5p is any

permutation, but always in the i th position, is a normal subgroup of the set of

all row-latin squares.

The following lemma is useful and obvious:

LEMMA 1. A row-latin square is orthogonal to the square I if and only if it

is a latin square.

It will also be helpful to have the lemma:

LEMMA 2. // A9 B9 ••• , L are a set of mutually orthogonal row-latin squares

and X is any row-latin square9 then XA, XB9 , XL are a set of mutually or-

thogonal row-latin squares.

GROUPS OF ORTHOGONAL ROW-LATIN SQUARES 337

To prove Lemma 2 it is sufficient to show that if A and B are orthogonal then

XA and XB are orthogonal. By Theorem 1 they are row-latin. If they are not or-

thogonal, the greco-latin square obtained by composing them contains some re-

peated number pair (u, v). Suppose a repeated pair occurs in row m, column p,

and in row n, column q. Let the element of X in row m, column p be x(m9 p ), and

similarly label the elements of A and B. Then

u = a[m, x{m, p ) ] = a[n, x(n, q)]

while

v = b [ m , x ( m , p ) ] = b [ n , x ( n 9 q ) ] .

So the greco-latin square composed from A and B contains the pair (u, v) in row

m, column x(m, p), and in row n, column x(n, p ). Since A is assumed orthogo-

nal to B this is a contradiction.

THEOREM 2. Two row-latin squares A and B are orthogonal if and only if

there is a latin square L such that AL = B.

If L is any latin square, then, since L is orthogonal to /, AL is orthogonal to

AI by Lemma 2; hence, A is orthogonal to B, Conversely, if A is orthogonal to

β, then by Theorem 1 there is a row-latin square L such that AL = B. We have

B ιA — L? B ιB = /; so, by Lemma 2, L is orthogonal to /. From Lemma 1, L

is latin.

If S is a member of a set of m mutually orthogonal row-latin squares, multiply

each square of the set on the left by S ι . The result, by Theorem 1 and Lemma

2, is still a set of m mutually orthogonal row-latin squares. Since it contains

5 ι 5 = /, all other squares of the set are orthogonal to / and are latin by Lemma

1. A complete set of mutually orthogonal latin squares may always be extended

as a set of orthogonal row-latin squares by adjoining the unit square /. Therefore

we have:

THEOREM 3. A row-latin square S is a member of a set of m mutually or-

thogonal row-latin squares if and only if there exists a set of m — 1 mutually

orthogonal latin-squares.

A set of n mutually orthogonal row-latin squares of order n will be called a

complete set. This is the maximum number of row-latin squares of order n which

can belong to a mutually orthogonal set. A set of n-\ mutually orthogonal latin

squares is customarily called a complete set of latin squares. The following

corollary is immediate from Theorem 3*.

3 3 8 DONALD A. NORTON

COROLLARY 3a. There exists a complete set of mutually orthogonal latin

squares if and only if each row-latin square is a member of a complete set of

mutually orthogonal row-latin squares. If such a set exists for one row-latin

square, it exists for every row-latin square of the same order.

3. Powers of A. In a row latin square A = (5βp , $βn ) each permutation

5βj has an exponent p(i)9 the least positive integer such that 5β?'1' = /. Let

p = LciD. [ p( 1 ) , . - . , p U ) L Ύhen AP = /, but 4? £ I for 0 < q < p.

The squares /, A, , AP~ι form a series of row-latin squares. If A is latin

then each is orthogonal to its predecessor in the series. Let m be the smallest

exponent such that Am is not latin. Then any m successive powers of A form a

mutually orthogonal set of row-latin squares. For suppose that the squares are

A 1 , , A i + m ~ ι . Ui<j<k<i + m-l, t h e n A k =-AUk~L S i n c e k - j < m ,

A J is latin; and A is orthogonal to A} by Theorem 2. Therefore we have:

THEOREM 4. If A is a latin square and m is the smallest exponent such that

Am is not latiny then any m successive powers of A form a set of mutually or-

thogonal row-latin squares.

The theorem of H. B. Mann [3, p. 418] follows as a corollary:

COROLLARY 4a. The squares A9 ••• , Am 1 are a set of mutually orthogo-

nal latin squares if and only if they are all latin squares.

We need the following:

THEOREM 5. If A is a latin square> then so is A~ι.

By Theorem 1, A~ι is row-latin. Since A is orthogonal to /, A ιA - I is or-

thogonal to A ι ' / = A ι by Lemma 2. Then by Lemma 1, A ι is latin.

Combining Theorems 4 and 5 we have:

COROLLARY 5a. // A, •• ,Am ι are latin squares, then any m-1 suc-

cessive squares of A~m ιA ~m 2 , , A ι

fA9A2

f 9 Am~ι form a tnutually

orthogonal set of latin squares.

Suppose Ap = /. Then A~J = AP~J, so we have the following:

COROLLARY 5b. // Ap = 1, and A, ••• , Am~ι are latin squares for some

m— 1 > p/2 then A9 , A? are a set of mutually orthogonal latin squares.

Examples may be constructed to show that it is not true conversely that

if A9 ••* 9 An are latin squares, and Ar, for some r > n, is a latin square, then

Aτ belongs to a series of n successive powers of A which form a mutually or-

GROUPS OF ORTHOGONAL ROW-LATIN SQUARES 3 3 9

thogonal set.

4. Squares as multiplication tables. A pseudo-latin square may be con-

sidered as the multiplication table of a groupoid. Two groupoids G( ) and H{°)

are isotopic if there exist mappings 11, 55, and SB of G into H such that

U y) 8 = Utt) o(yB)

for all x and y of G (for this and other concepts for finite multiplicative systems

see for instance Bruck [ l , pp. 245-255] ). If the groupoids are defined on the

same set G, then the mappings 11, 55, and 2$ induce a permutation of the rows,

columns, and elements respectively of the multiplication table of G( ), trans-

forming it into the multiplication table of £ ( ° ) . It is natural therefore to call

two pseudo-latin squares isotopic if one may be transformed into the other by

a permutation of rows, columns, and elements. The row-latin, column-latin, or

latin squares are then multiplication tables of groupoids in which every element

is left nonsingular, right nonsingular, or nonsingular, respectively.

Every latin square is isotopic to a standard latin square in which the first

row and first column are the elements in their natural order. A latin square is a

basis square [3 p. 249] if there is a latin square orthogonal to it. If A is a basis

latin square, there are latin squares X and B such that AX = δ . From this we

have X - A~ιB, and A'1 is also a basis square. If A = A ι then A2 = /. All

other basis latin squares occur in pairs A and A ι , so there are an even number

of basis latin squares with exponent greater than 2.

Let A be an n x n row-latin square of exponent 2. If α, 6, c, are the

elements of the groupoid defined by A, then

(1) bll = b .

Let pirn) be the number of ways of selecting the order of m elements of row a of

A to satisfy (1). Equivalent to (1) is the statement that ab = c implies ac = b. If

c = 6, the element of the 6 th column alone is determined. If c ^ b the elements

of both columns b and c are determined, so only n — 2 remain to be fixed. Hence

p{n> = 1 p<* - 1> + U - 1) p<n - 2> .

Since each row is independent of the other rows, we have:

THEOREM 6. The number of n x n row-latin squares with exponent 2 is

[p(n)]n, where pin} is given inductively by

p(n> = p < n - l > + U - l ) p < n - 2 >

and

3 4 0 DONALD A. NORTON

p<l>= 1, p<2> = 2.

If the square is further restricted to be standard, the first row is prede-

termined as well as the first element of each row. But if this first element is 6,

then in the statement equivalent to (1) above we have c ^ b so that two elements

of each row are predetermined and we have:

COROLLARY 6a. The number of n x n standard row-latin squares with

exponent 2 is [pin - 2 > ] n " 1 .

We might further note that p<2> = 2, p<3> = 4, so that the number o f / i x / i

row-latin squares with exponent 2 is even for n > 1.

Suppose A is a standard latin square with exponent 2. It is the multiplication

table for a loop, and the element in row i, column j i s the product element ij in

the loop. The permutation §βj carries the element j into the element ij. Repeating

5βj further carries ij into i(ij) - jL2; so the element of the ith row, /th column

of A2 i s jL2. Since A2 = /, then jL2 = / for all i, /'. In particular if j = i then

/ ( / 2 ) = /, so j 2 = 1 for all y. Every element of the loop has exponent 2, and the

loop has the left inverse property. Conversely if the loop has the left inverse

property and every element has exponent 2, then i(ij) - i~ι(ij) = 1 y = y.

Therefore A2 = /. We have proved the following:

THEOREM 7. If A is a standard latin square, then A2 - I if and only if the

loop defined by A has exponent 2 and has the left inverse property.

If A is an n x n latin square with exponent p, let {A} be the cyclic group

of elements /, A, ••• , Ap~ι. If { A \ is a set of mutually orthogonal row-latin

squares then p < n.

As before, if A i s considered as the multiplication table of a loop, the ele-

ment in the £ th row, yth column of AΓ i s y'L£. Then jLV = y. If {A \ i s a set of

mutually orthogonal row-latin squares, then A, ••• , Ap~ι are latin squares; so

for any r < p and for any j , we have jL7^ = jU^ if and only if i = k. This proves:

THEOREM 8. If A is a row-latin square with exponent p, then \ A] is a group

of mutually orthogonal row-latin squares if and only if for any /, we have jL? -

j, but for any r < p the equality jL^ = jL^ implies i = k.

If a finite loop of order n has a subloop of order m < n9 then it contains an

element i with left exponent p, 1 * L? = 1, p < m* If A is the multiplication

table of the loop then Ap is not a latin square. So we have:

COROLLARY 8a. If A is a latin square, then a necessary condition that Ap

be a latin square is that the quasigroup L, defined by Af be not isotopic to a

DONALD A. NORTON 3 4 1

loop with a subloop of order less than or equal to p.

REFERENCES

1. R. H. Bruck, Contributions to the theory of loops, Trans. Amer, Math. Soc. ΘO(1946), 245-354.

2. H. B. Mann, The construction of orthogonal latin squares, Ann. Math. Statistics13 (1942), 418-423.

3. , On orthogonal latin squares, Bull. Amer, Math. Soc. 50, 249-257.

4. L. Paige, Complete mappings of groups, Pac. J. Math. 1 (1951), 111-116.

UNIVERSITY OF CALIFORNIA, DAVIS

ON THE GENERATION OF SEMIGROUPS OF LINEAR OPERATORS

R. S. P H I L L I P S

1. Introduction. Let T(ξ) be a semigroup of linear bounded transformations

on a Banach space 3C to itself (see [4]), strongly continuous on [0, oo) with

Γ(0) = /. Further let 0C( ί, ξ) be a one-parameter family of functions of bounded

variation in ξ >^ 0 which form a semigroup in t >^ 0 with product defined by

convolution. Then

(1) S ( ί ) = JΓ°° T(ξ) dξ a(t,ξ)

will also form a semigroup of linear bounded transformations on 3C to itself. If

the semigroup 0C(ί, •) satisfies certain continuity conditions, then S(t) will

be strongly continuous for t > 0. This method of generating new semigroups

out of old ones has previously been considered in a general way by N. P.

Romanoff [10] and in connection with stochastic processes by S. Bochner [2].

We shall consider the problem in the setting described above and attempt to

obtain the infinitesimal generator B of the semigroup S ( ί ) directly in terms

of the semigroup T{ξ) and its infinitesimal generator A. We shall seek also to

relate the spectrum of B to that of A

In general the integral in (1) will not converge unless 0ί is suitably re-

stricted. The most general function (X of bounded variation, for which the inte-

gral converges absolutely, will satisfy the condition

( 2 ) JΓ°° exp [ * > ( £ ) ] \da\ < oc,

where ω(ζ) = log | l ^ ( f ) | | is lower semicontinuous, subadditive, and bounded

near the origin. We shall accordingly limit ourselves to the Banach algebra

G(α>) consisting of the set of all such functions α, where the norm is given by

the integral in (2) and the product is given by convolution. Now S ( ω ) can also

be considered as an operator algebra over the Banach space C(ω) consisting

of the absolutely continuous elements in Θ(ω). In the course of proving that

the operator topology forS(ω) and the original topology are isomorphic we have


343

344 R. S. PHILLIPS

obtained the following relation for the case lim ω(ξ) = 0:£-0

(3) j Γ exp[ω(£)] | rfα | = lim /o°° exp dξ.

We have been able to extend the results of A. Kolmogoroff [7] and Paul Levy

[β] on semigroups of distribution functions to the case of semigroups of mono-

tonic nondecreasing functions in 6(ω) when ω(ζ) satisfies the additional

condition

(4) lim / s u p [ω(ξ) - ω(ζ + 8)]) < oo.δ-° \S 1 o /

In fact, if we set ωo = inf ω (ζ) / ζ and

(5) Φ[λ, α] = fQ°° e x p [ λ ^ ] ^ α for R [λ] < ω0 ,

then we have shown that a necessary and sufficient condition that 0C(ί, ξ) be

a semigroup of monotone functions in G(ω) (strongly continuous over Q(ω) at

t = 0) is that

(6) Γ 1 logΦ[λ, α(ί, •)] = mλ + fj0 ( e x p [ ( λ - ω o ) f ] - 1) dφ + α,

where m > 0, a is real, and φ is monotonic nondecreasing on (0, oo) satisfying

the conditions

J ζdφ < oo and J exp [ω(£) - ω0 ξ] c?ψ < oo.

Finally, we have been able to obtain a characterization of the infinitesimal

generator B of S(t) when S(t) is generated by a semigroup of monotone func-

tions in G(ω). In this case we show that for x in the domain of A we have

(7) Bx = mAx + f°° [exp(-ωo£) T(ξ)x - x] dφ + ax,

where m, α, and φ(ζ) are obtained from (6). In addition, if λ0 belongs to the

spectrum of A, then

77iλ0 + f°° (exp[λ0 - ωo)ξ] ~ 1) dφ + a

belongs to the spectrum of B.

ON THE GENERATION OF SEMIGROUPS OF LINEAR OPERATORS 345

2. The Banach algebra 6 ( ω ) . This section will be devoted to a character-

ization of various kinds of convergence in the Banach algebra δ ( ω ) . We shall

suppose that the weight function ω(ζ) satisfies the hypothesis

( i ) ω(ζ) is real valued and lower semicontinuous on [0, oo );

(h) ( i i) ω ( 0 ) = 0, Thn" ω(ξ) < oo;

(iii) ω(ξ) is suhaά&tive: ω(ξχ+ξ2) < 0)^) +ω(ξ2) toe ξιf ξ2 > 0.

If, in a d d i t i o n ,

( i v ) lim f s u p (ω{ξ) - ω(ξ + δ ) ) l < oo,

we shall say that ω satisfies the hypothesis (A*). Conditions ( i ) and ( i i i )

imply ( see [4, Chap. 6]) that lim* + ω(ξ) _> 0, and hence that ω{ξ) is

bounded in every finite interval [0, L ]. Further,

(8) ω0 = inf ω(ξ)/ξ= lim ω(ξ)/ξ.ξ > o ξ-+<*>

We now define δ ( ω ) to be the set of all completely additive complex-valued

set functions CC(σ) on the sigma-field of Borel measurable subsets B of [0, oo)

such that

/°° exp [ω(ξ)]\dα\ < oo.

We require only that CC(σ) be finite if σ is contained in a finite interval. The

norm is given by

(9) | | α | | = ^°° e χ p [

It is clear that S ( ω ) is a Banach space. In order to define the product γ = (X *j3

of two elements of 6 ( ω ) , we consider the product measure γ of CC and β de-

fined on the smallest sigma-field S 2 generated by ϊ x 35. For any σ £ B,

we set

γ(σ) = ψ[(u, v) I u + v C σ; u, v >^ 0 ] .

It follows that the product is commutative, that

( 1 0 ) γ ( σ ) = / 0 L ( σ - u ) d u β = J β(σ-u)du(X,

346 R. S. PHILLIPS

and that

(ID l l y l l <

Thus S(α>) is likewise a Banach algebra with unit e. It will sometimes be con-

venient to consider 0 t€ S(ω) not as a set function CX(σ), but simply as a

function of bounded variation continuous on the left, namely as

This correspondence between set function and point function is clearly one-to-

one and should not cause any confusion.

Let Q (ω) be the set of Borel measurable functions f(ζ) such that

fo° f{ξ)

Then with norm

(12) | | / | | = J f « p [ ω ( £ ) ] \f(ξ)\dξ,

Q(ω) is a Banach space Further if α G δ(ω), then

(13) Aa(f) f(ξ - u) dua

defines a linear bounded transformation on Q(ω) to itself. Clearly

I K ( / ) I I < | | α | | 11/11,

and hence

(14) | |Λ«|| < | | α | | .

We shall show that the two norms are isomorphic. To this end we set

1/δ for 0 < ζ £ 8 and zero elsewhere. Then by (14),

(15) lim f°° exp[ω(f)]-•0+ °

0-* 0 +

dξ

\ A Λ ( i % ) \ \ < c\\a\\,

where

(16) c = lim δ~ ι

s - o +

Because of (h-ii) and the subsequent remarks, it is clear that 1 £ c < oo.

LEMMA 2.1. If ω(ξ) is subadditive and merely lower semicontinuous, then

(17) lim Γ exp[ω(£)]- δ)

dξ > \\a\\.

Since exp[ω(ζ)] is lower semicontinuous, we can approximate it from below

by a sequence of continuous functions wn(ζ) such that

and lim wn(ζ)n—*oo

pointwise. We shall show that

(18) lira Γ wn(ξ)8-0+ °

- α ( ^ - δ)dξ>

and hence that

lim Γ- α ( ^ - δ)

for all n. The inequality (17) then follows by Fatou's lemma. It remains to prove

(18). Now given e > 0 there exists a subdivision ξQ = 0 < ξχ < < ξn ~ L

such that

(19) u,(ξf)

where ^Γ

ί_1 £ '? < ξ^ and this inequality remains true for all refinements of

the above subdivision of [ 0, L] Setting

du,

we have

348 R. S. PHILLIPS

j* w{ξ)[0i{ξ)- OL(ξ-δ)]dξ

- fb W(ξ)da(ξ) + fb6

la "a

(20)

[ W { b ) ~ - fa [W(a) - δ)]da

[W(ξ + 8)-W(ξ)]da.

Now

δ) - W(ξ)]dθL- w(ξ')δ[a(b) - α ( σ ) ] |

£ O S C [ M ; ; α , b + 8] v a r [ ( X ; α , b]f

where a < ξ' <_ b. If M = max [ | κ> ( f ) | | f £ [ θ , L ] ] , we see that the first two

terms on the right of (20) are bounded above by Mδ var[θC; b — δ, b] and by

M8 var[(X; a - δ, a] respectively. Hence

%-ι

- a(ξ~ δ)

> ΣS

> Σ ^(^') lαίf^-

osc[u;; f..^ ^. + δ] var[α; ξ^χ9 ξ.

- i f a ' ^ - i -δ, f..1] + var[α;f. -δ,ί f

We first choose a subdivision sufficiently fine so that for all sufficiently small

δ > 0 we have

ON THE GENERATION OF SEMIGROUPS OF LINEAR OPERATORS 3 4 9

max \osc[w; ξimmχ9 ξ. + δ]} <v a r [ α ; 0, L]

Combining (19) with the above inequality we then get

JP00 t t\ \ J i C°° / Λ x ^ vs / *" tt(f - δ)

n

+ 2e + 2M ^ var[α; ξ. - S, ξ.]

1 = 0

The inequality (18) now follows from the fact that 0L is continuous on the left

and the arbitrariness of e > 0.

As a consequence of Lemma 2.1 we see that

P a l i = sup | K ( / ) | | / 11/11 > Hm P a ( / 8 ) | | / Thί | | / s | | > | | a | | / cδ-»o+ δ—o+ °

Combining this with (14) we have

(21) | |α||/c < p α | | <

THEOREM 2.1. The operator norm for the elements of G(ω) over Q ( ω )

is isomorphic to the regular norm (9).

In particular, if lim,. + ω(<f) = 0 then c = 1, and hence (3) follows from

(15) and (17).

We next obtain a criterion for the strong convergence of elements in G(ω)

considered as operators on Q ( ω ) .

THEOREM 2.2. The operators Aa converge strongly to 0 if and only if the

110tn 11 are bounded and

(22) lim j Γ expU(<f)] | 0Ln (ξ) - an (ξ - δ) | dξ = 0

for all sufficiently small δ > 0.

K "mπ_*oo Aan (/) = 0 for each / C ? ( ω ) , then by a well-known theorem due

to Banach [ l , p. 80] the | | / 4 α n | | are bounded, and hence by Theorem 2.1 so are

the | |θC n | | . Further, if f§(ζ) = 1 for 0 < ξ £ δ and zero elsewhere, then

3 5 0 R. S. PHILLIPS

IMαB(/S)ll = j f

for all δ > 0. Conversely, suppose (22) is valid for all δ C (0, δ0 ). Then given

a > 0 there exists an integer k and a δ C (0, δ0 ) such that A δ = α. Now fora n v ζQ — 0 we have

0 j - 0Cn (ξ - 8) \dξ

as n—»oo. Hence

^ Σ

as n—>oo. In other words, if / ( f ) = 1 for 0 < ζ < a and zero elsewhere, then

I M α r a ( / ) | | — * 0 . Since the linear extension of this class of functions is dense

in C(ω), and since |Mαn II a r e bounded by (14), the desired result follows from

the Banach-Steinhaus theorem [l, p. 79].

We now define

Φ(λ, α) = f°° exp[λξ]d(X,

(23)

/)= Jo°° ex?[λξ]f{ξ)dξ.

These integrals converge absolutely for R[λ] £ ω 0 . We recall that the Laplace

transform of the convolution of two functions is the product of their Laplace

transforms, and hence that

(24) f" exp[λf] Aa(f)dξ=Φ{λ)φ{X).

We next obtain a necessary condition for the strong convergence of the operators

Aan, where again an G 6(ω).

THEOREM 2.3. If the operators Aa converge to 0 in the strong topology,

then the | |θCn | | are bounded, Φπ(λ)—»0 uniformly in every bounded subset of

R[λ] £ ω0, and Oίn(ξ)—>0 pointwise.

The boundedness of the | | θ C Λ | | follows as in Theorem 2.2. Since

\φ(Kf)\ < 11/11 *<* R [ λ ] < ω 0 ,

it follows from (24) that | Φn{λ) φ(λ, f) \ < | | 4χn(/) | | -» 0 uniformly for

H[λ] £ ωo The uniform convergence of ΦΛ(λ) to zero on a bounded subset

E of R[λ] £ ω0 now follows from the fact that there exist functions φ(λ, f)

bounded away from zero on E. In fact, for /g defined as in the previous theorem,

φ{\, fs )= ( e x p ( λ δ ) - l ) / λ will suffice for δ sufficiently small. The fact

that &n(ξ)—»0 is a consequence of the following lemma, patterned after the

P. Levy convergence theorem [8, p. 49].

LEMMA 2.2. // an C S(ω), the | |θCπ | | are bounded, and Φ(λ, an) con-

verges to Ω(λ) uniformly in every bounded subset of R[λ] < ω0, then there

exists an α C S ( ω ) such that Φ(λ, Oί) = Ω(λ) and OLn(ξ)—»α(£) at every

point of continuity of α.

Since the variations of the CLn are uniformly bounded in every finite interval,

we can apply the Helly theorem and obtain a subsequence CLnk{ξ) which con-

verges to a function OC (ξ) (likewise of bounded variation in every finite interval

and continuous on the left) at each point of continuity of 0t(£). In order to show

that OtC δ(ω), we approximate exp[ω(^)] as in Lemma 2.1 by a sequence

of continuous functions wn(ζ)9 where

0 < wn{ξ) < αfc+ ι(£) and wn(ξ)—

pointwise. Then, by hypothesis,

JH° ™n(O \dank\ < jΓ°° exp[ω(ξ)] \dθink\ < M.

For each L > 0 we have

fL wn(ξ) \da\ < lim ΪL Wntf) \dCtnk\.

Hence J wn(ξ) \dθL \ <^ M for each n; and, by Fatou's lemma,

f°° exp[ω(ξ)] \da\ < M.

352 R. S. PHILLIPS

Thus α C 6 ( ω ) , On the other hand, for any δ > 0 and u = R (λ ) <. ω 0, we have

( 2 S Γ 1 fV*S Φn(u + iv)</i/ = / ~ exp[(u + iv)£] (δξ)~ι sin δ£ rfαn.f "~ o 0

Φn(u + iv)</i/ /f "~ o 0

Now

1 °° exp[(α + iv)δ] (δξΓ1 sin δξ dan\ < M/(SL)9

so that

J °° exp[(u+ «;)£] ( δ f Γ l s i n δ f ^ α n

—» J°° exp[(a + iϋ)^] (δξ)~ι s inδf rfα.

Finally, since Φn (λ)—> Ω(λ) uniformly in every bounded subset of R[λ] < ω 0,

we obtain, for all δ > 0,

( 2 δ Γ ι j Γ * 8 Ω(u + iv)dv = jΓ°° exp[(« + iv)ξ] (8ξ)~ι sinδξdΰL

= (28)~ι Γ + S Φ ( B + iv)dv.Vm" o

Thus Φ(λ) s Ω(λ) for R[λ] £ ω 0. Finally since this is true of all subse-

sequences, it follows from the uniqueness theorem for Laplace transforms that

>OL(ξ) at all points of continuity of

3. Nonnegative semigroups in S ( ω ) . In this section we shall obtain a gener-

alization of the Kolmogoroff [7] and P. Levy [8, Chap. 7] representation for

semigroups of distribution functions in terms of their characteristic functions. We

shall consider the semigroup α ( ί , ξ), where

( i ) for each t >^ 0, 0C (t, ξ) is a nondecreasing function in 6 ( ω ) ;

( ϋ ) α(tx + t2, . ) = α ( ί l f •)* α ( ί 2 , •), α ( 0 , •)= β;

(25) ( i ϋ ) there exists an M such that | | α ( ί , ) | | < M for 0 < t < 1;

(iv) Φ [ λ ; α ( ί , •)] = f°° exp (λf)rf^ α ( ί , ^) converges to one as ί—->0+funiformly in any bounded subset o f R [ λ ] _< ω o

It will be convenient to introduce the auxiliary space S ( ω ) , where

ω (ξ) = ω(ξ) - ωoξ.

It is clear that ω(ζ) will also satisfy the hypothesis (h) and that inf ω(ξ)/ξ =

0. Hence if β £ G(ω), then β is a function of bounded variation on [0, oo).

The transformation

(26) β ( σ ) = U ( α ) ( σ ) s £eiap(

is a norm preserving isomorphism of the Banach algebra S ( ω ) onto 6 ( ω ) .

Further,

Φ(λ, α) = JΓ°° ejφ{\ξ)dξa = JΓ°° exp[{λ- ωo)ξ]dξβ = Φ(λ- ω0, β).

Hence if

β(t, . ) = U [ α ( ί , •)]

then all of the conditions (25) are fulfilled for the semigroup β(t, ) C S ( ω )

with 0 replacing ω0 in (iv). Since, in particular,

O < j 8 ( t , o o ) < | | j 8 U . ) | | < J l ί forO < ί < 1,

β(h + t2, oo) = β(h, oo) β(t2, oo),

it follows from well-known results on multiplicative functions that β(t,co) =

exp(αί) We now define

(27) γ(t,ξ) = e x p ( - α f ) β(t9ξ).

Then y( ί , ) is a semigroup of distributions, continuous in the sense of (iv).

Hence we can apply the results of Kolmogoroff and P Levy, We refer the reader

to Khintchine's proof [6] which has been reprinted by Hille [4, p. 435]. Implicit

in P. Levy's discussion of this theorem [8, p. 178] is the following modification,

valid for one-sided distributions (that is, distributions defined on [0, co))

LEMMA 3.1. Let γ(t$ ξ) be a semigroup of one-sided distributions such that

(28) lim y(ί , ξ) = 1 for each ξ > 0.ί-» o +

Then

(29) Γ ι log(Φ(λ, y(ί, •)]) = ™λ + j Γ t e x p U a - l]dψ(ξ)

for all R(λ) £ 0, where m > 0; φ(ζ) is nondecreasing on (0, oo); φ (<x>) < oo;

and J ξ dφ < oo. Conversely, every such choice of m and φ defines a semi-

group of onesided distributions satisfying (28).

For the sake of completeness, we shall sketch a proof of this fact. The

reader will be able to fill in the details by referring to Hille [4, pp. 435-438].

It is readily seen that

log(Φ[λ,yU, •)]) ^ tΦχ(λ),

and hence that

(30) Φi(λ) = lim Γ ι f°° [exp(λf) - l ] dμ γ{t9 ξ).

We next define

G U ξ) = Γ ι j£ f 7/2/(1 + 772) dv y(ί, 77).

As in [4], it can be shown that var[G(ί, •)] is bounded for 0 < t < tQ; that the

variation of G(t, ξ) in [L, 00) goes to zero as L—>oo, uniformly for 0 < t _< ίo;

and that for some sequence tn tending to 0+,

G{tn,ξ)—>G(ξ) and mn = jΓ°° ξ~ι dG(tn, ξ) —> m0 > 0.

It follows that G(ξ) can have no jump at the origin and that

WQ ^ lim I ζ dG ( t n 9 ζ) = I ζ

Thus jΓ°° ξ~ι dG(ξ) < oo. Hence, by (30),

Φ t(λ) = lim J mnλ71 — oo I

ξ2)/ξ2] dG(tn,

= mλ+ fo°°[exV(kξ)-l]dφ(ξ),

where

m = m 0 - £°ξ~ιdG(ξ) > 0 and φ(ξ) = - fg°° {l + η*)/η* dG{η).

Thus the function φ{ζ) is monotonic nondecreasing on (0, oo); φ(ζ) <_ 0;

and J ξ dφ(ξ) < oo. The uniqueness of φ(ζ) follows from the uniqueness of

G(ζ) as in [4]; the converse statement is also proved as in [4]

For our purposes we shall need a theorem of this type applicable to semi-

groups in G(α7) satisfying the conditions of (25). We shall show that if

f exp[ω(ξ)] dψ(ξ)<

then y(ί, ζ) G G(ω), and that the converse is likewise true for suitable re-

stricted ω(ζ). Without loss of generality we set t = 1.

THEOREM 3.1. //

(31) logΦ(λ,y) = mλ+ j H [ e x p ( λ f ) - l]dψ(ξ),

where

m > 0, fl ξ dφ < oo, and f°° exp[ω(£)] dφ(ξ) < oo,

then γ £ S ( ω ) .

It is clear that y is the convolution of three one-sided distributions, namely:

γ = an m-shift to the right,

7 2 , where log Φ ( λ , y 2 ) = £ [exp (λξ) - l ] dφ,

y3 , where logΦ(λ, y3 ) = ^° ° [exp(λf)- l ] dφ.

It is further clear that γχ C 6 (ω), so that it remains to show that the same is

true of y2 and y3 Now J* [exp(λ^) - 1] dφ(ξ) can be approximated by sums

of the type

uniformly in every bounded subset of R(λ) <_ 0. Suppose y is such that

logΦ(λ,y7r)= £

356 R. s. PHILLIPS

Then γ^ is the convolution of n Poisson distributions, and hence has a jump of

n n ( Δ J T / Γ ) * n

" ^ Zmί iΨ' L i L#] Z* i ^i *

It follows that

( n I n \ n {ΔiXlj)ki

Now by assumption (h-ii) there exists an M > 0 such that exp[ω( ξ)] £ M for

0 £ <f £ 2. Hence, in general, exp[ω(£)] £ M^ for all ξ >_ 1. Thus we obtain

1 = 1

Finally, since f exp[ω(^)] dyπ(ξ) £ Λf, and since ^* (M* — \) dψ < oo,

the approximating γπ*s can be chosen to satisfy the conditions of Lemma 2.2.

It follows that y2 C S(ω). Likewise J°° [exp (λξ) - 1 ] c?0 can be approxi-

mated by sums of the type

ί = l

uniformly in every finite subset of R(λ) £ 0. Hence y3 can also be approxi-

mated by distributions of the type γπ. Now

« | Σ *<£•) l Σ *iω(^ ),

\ί-I / «-I

so that in this case we have

nl og||yπll £ Σ [ e x p ω ( ^ ) - l ] Δ ^ .

1 = 1

Since J (exp[ω(f)] - 1) dip < oo, it follows as above that y3 C S(ω).

THEOREM 3.2. If γ{t, ξ) is a semigroup of distributions in G(ω) satisfying

the conditions (25), and

(32) lim sup [ω (ξ) - ω( ξ + 8)] < oo,

then

Γ 1 log Φ[λ, y(ί, )] = mλ + JΓ°° [exp(λ£) - I] dφ ,

where m >_ 0, φ{ζ) is monotone nondecreasing with

J ξdφ < oo, and J°° exp[ω(f)] dφ < co.

It is clear that the continuity conditions (iii) and (iv) of (25) imply (28) by

Lemma 2.2. Hence we may avail ourselves of the results of Lemma 3.1. It re-

mains therefore only to verify the statement J exp [ ω ( ξ ) ] dφ < oo. By the

assumption (32), there exists a Δ > 0 and a k > 0 such that ω(ξ + 8) >. ω{ζ)- k

for all ξ >_ 0 and all 0 < 8 £ Δ. It follows by an induction argument that

ω ( f + δ) > ω(ξ)~ (δΔ~ ι + l)k for all ξ _> 0 and all δ > 0. Suppose now that

γ( ξ) is of bounded variation on [0, oo) and that γ(ξ- m) £ G ( ω ) . Then

I dγ I < exp[(mΔ 1 + 1)A;] / e x p [ ω ( £ + m)υ ''0

so that γ{ξ) £ G(ω). Thus without loss of generality we may assume that

m = 0 For t equal to, say, one, we define γ and y as before, so that2 3

logΦ(λ,y2) = £

and

l o g Φ ( λ , y 3 ) = . f

Then

Π ll II II / °° / c

y ! l = l l y 2 * y 3 l l = X X

358 R. S. PHILLIPS

whence

Since y2 is a distribution function, we have y2 ( e ) > 0 for some e > 0. Hence

y 3 C δ ( ω ) . Finally we see that

Φ(λ,y3) = exp[./ i

θ°#] £(Φ(λ,y 4))n

n\\

where y4( ξ) = 0 for 0 < ξ < 1 and y4 ( ξ) = φ (ξ) - φ (1) for ξ > 1. In other

words,

and in particular y3 (σ) > exp[ — J* c?^3 y4 (σ). Since y3 C S(ω), it follows

that y4 G S(ά7), and hence that

J exγ>[ω(ξ)]dφ < oo.

We summarize the results of this section in the following theorem.

THEOREM 3.3. // 0C(ί, ξ) is a semigroup of elements in 6 ( ω ) satisfying

the condition (25), and ω(ξ) satisfies (A*), then for R(λ) £ ωo we have

(33) Γ ι logΦ[λ,α(ί, •)] = rnλ + jΓ β ° (exp[(λ-ω 0 ) f ]- l )^(έ : ) + β»

where m >_ Q, a is real, and φ(ζ) is a monotone nondecreasing function such

that

f ζdφ < oo and J exp[ω(£) - ωoξ] dφ < oo.

Conversely, if Φ[λ, 0C(ί, •)] satisfies (33), ίλeπ α(ί, •) G S(ω) and satisfies

(25) for any ω(ξ) satisfying (h).

Before we conclude this section, a remark is in order about the continuity

of the semigroup α(ί, ). If we consider /4 ,£ v to be a semigroup of operators

on Q (ω), then the strong convergence of these operators to the identity operator


as t—> 0+ implies, by Theorem 2.3, that the continuity hypothesis (25)- (in)

and - (iv) will be satisfied by 0ί(ί, )• We shall show in Corollary 4.1 that the

converse is likewise true.

4. Semigroups of transformations. In this section, we shall make use of

the Banach algebra 6 ( ω ) to develop an operational calculus for the infinitesi-

mal generators of semigroups of transformations on a Banach space to itself.

Hille [4, Chap. 15] first introduced such a calculus. The novel feature of the

present discussion is that this calculus is used to obtain other semigroups of

transformations, and as a consequence to obtain other infinitesimal generators.

This method of generating new semigroups has previously been considered in a

general way by N. P. Romanoff [lO] and in connection with stochastic processes

by S. Bochner [2].

Let X be a complex Banach space, let (§r( X) be the algebra of linear bounded

transformations on X to itself, and let T(t) be a semigroup of operators on

[0, oo) to S(X) (see [4]) satisfying the following hypothesis:

(H)( i ) T ( t x + t2) = T{tx)T{t2) f o r * ! , t2 > 0, Π 0 ) = /;

(ii) lim T{t)x = x for all % G X.

Such a semigroup of transformations will have a closed linear infinitesimal

generator A with domain 2) (/4) dense in X. It can be shown [4, Theorem 9.4.1]

that T{t) is then strongly continuous for t >. 0, and hence that | | ! Γ ( £ ) | | * s

lower semicontinuous. It is clear that the subadditive function

(34) ω(£) = l o g | | Γ ( £ ) | |

satisfies the conditions (h) and may be used to define a Banach algebra of the

type S(ω). For 0ί(£) C 6(ω), the relation

(35) Θ(α)*Ξ f°°T(ξ)xdθL(ξ)

defines a linear bounded transformation in S(3C). Further, it is easily seen that

Θ ( e ) = /,

(36)Θ(α * β) = Θ ( α ) Θ ( / 3 ) ,

Hence Θ(θC) is a continuous homeomorphism of 6 ( ω ) into S(3£) which takes

the unit e into the identity /. This mapping can be thought of as defining an

operational calculus for the infinitesimal generator A of T(t) (see [4] and [9]).

Suppose now that d(t9 •) £ 6 ( ω ) form a semigroup of set functions such

that A»t v considered as operators on Q (ω) satisfy the postulates (H) Then

(37) S(t) = θ [ α ( ί , . ) ]

is clearly a semigroup of operators on 3£ to itself by (36). We show next that

S(t) converges strongly to / as t —» 0+. For this purpose we need the following

lemma.

LEMMA 4.1 Let Aan be a sequence of operators on Q(ω) which converge

strongly to O Suppose G(ζ) £ Q (ω)* is such that

(38) lim sup \G(ξ+ 8) - G{ξ)\ e x p [ - ω ( f ) ] - 0.g_»0+ ξ >_0

Then

(39) lim f°°G(ξ)dan(ξ)- 0.7Ϊ-K5O 0

By hypothesis,

πHm /o~ G{ξ) [fj f(ξ- u)dθin(

for each / £ ? ( ω ) . By the Fubini theorem,

If we choose /g (ξ) = δ""1 for 0 £ ξ <_ 8 and = 0 elsewhere, then

for each 8 > Oasn —> oo. On the other hand, by (38),

δ " 1 fu

U+SG{ξ)dξ- C ( u ) | < e x p [ ω ( u ) ] e ( δ ) ,

where €(δ) —>0 with δ . Hence

αnί.O- jfGUWMa)! < llαjl

which converges to zero as δ —» 0 uniformly in n. It follows that the order of

limits may be inverted and hence that (39) is valid.

It is clear that G(ξ) = x*[T (ξ)x~\ satisfies the condition (38) and belongs

to Q (ω)*. Further, by hypothesis, the A,t . \ converge strongly o n G ( ω ) t o

Ae = / as t —> 0+. Hence, by the lemma,

lim * * [ S ( ί ) * l - Hm f°° x*[T(ξ)x]da(t, ξ) = * * ( * )ί - o + t-κ>+ J°

for all Λ; £ X and #* £ X*. The desired strong convergence now follows by a

theorem due to Hille [5, p. 93, footnote]. This concludes the proof of the follow-

ing result.

THEOREM 4 . 1 . // T(t) is a semigroup of operators on X, and CX(ί, ) C

6 ( ω ) is a semigroup of operators on Q ( ω ) ( ω ( ζ) = log | j T{ ξ) \\ ), both satis-

fying the postulates ( H ) , then S ( ί ) = Θ [ θ C ( ί , •)] *>s again a semigroup of oper-

ators on X satisfying ( H ) .

If we limit ourselves to semigroups in S ( ω ) of the type studied in §3, we

are then able to obtain a representation for the infinitesimal generator of the

semigroup S ( ί ) = Θ[(X(ί, •)!• To this end, we now prove a generalization of

a theorem due to K. Yosida [ l l , Lemma 2].

THEOREM 4.2. Let 3ϊ be a strongly closed abelian subalgebra of ( S ( X ) .

For each integer n, let Sn(t) be a semigroup of transformations in 3ϊ satisfying

( H ) with infinitesimal generator Bn. We further assume that lim^^cxj ^n(x) =

B'(x) for a dense subset 2) C Γ\?)(Bn)9 and that there exists an M < oo such

that \\Sn(t)\\ £ M for t C [0 , 1 ] and all n. Then the Sn(t) converge strongly

to a semigroup of transformations S(t) in 3ί satisfying ( H ) with infinitesimal

generator B D B'.

In the proof of this theorem we make use of a technical device due to Dunford

and Segal [3]. For x C 2) C Π 5)( Bn ) it is easy to verify that Sn {t - Ί)Sm (Ί)x

is strongly differentiable with respect to T. Hence

Sm(t)x -Sn(t)x= J* i - [Sn(t- τ)Sm(τ)χ]dτ

= ίtSn{t-τ)Sm(τ)[BmX- Bnx] dτ.

I t f o l l o w s f rom o u r h y p o t h e s i s t h a t \\Sn (t)\\ £ M ι + ί , a n d t h u s t h a t

Since the Bn(x) form a Cauchy sequence, so do the Sn (t)x. We define the limit

to be S(t)x. Now for x C 2), S(t)x is the uniform limit of continuous functions

in every finite interval, and hence is itself continuous. Further, since 2) {s

dense in 3C, and because of the uniform boundedness of the | | S Λ ( ί ) | | for each

t, it follows that Sn(t)x —>S(t)x for all x G 3C and that | | S ( ί ) | | < Mιn.

It is a simple matter to verify that the S ( t ) form a semigroup of transformations.

Also S(t)x is strongly continuous on the dense set 2), for t >. 0 and | | S ( ί ) | | <

Uι ι implies that S(t) is strongly continuous on X for t >_ 0. Thus S(t) satis-

fies (H). Finally, f or x G 2) we have Sn(t)x = x + / * Sn{ξ)Bnxdξ and

Sn(ξ)Bnx —> S{ξ) B'x pointwise and boundedly. Hence

S ( f ) * = x+ JQ

tS(ξ)B'x dξ.

Thus, for x G D we have d(S(t)x)/dt = B'xy and hence 2) C 2)(β) and

Bx = B'x for* G S>.

We shall hereafter consider only semigroups of functions 0C(ί, ξ) (of bounded

variation in every finite interval) whose Laplace transforms Φ[λ, 0C(ί, )1

take the form

(40) ί"1 logΦ[λ, α(ί, •)] = mλ + jΓ°°(exp[(λ- ωo)ξ] - l)dφ(ξ)+ a

(R(λ) < ω0),

where m > 0, a is real, and φ(ζ) is a monotone nondecreasing function such

that

J ζdψ < co and J* exp[ω(f) — ω0 f] dψ < oo.

By Theorem 3.3 we see that α(ί, •) G δ ( ω ) and satisfies (25). The converse

will likewise be valid if ω(ζ) satisfies (32) in addition to the conditions (h).

THEOREM 4.3. Suppose that T(ξ) is a semigroup of transformations satis-

fying (H), that ω(ξ) = log | | Γ ( £ ) | | , and that α ( ί , •) is a semigroup of ele-

ments in 6 ( ω ) having Laplace transforms of the type (40). Then the semigroup

of transformations S(ί) = Θ[cx(ί, •)] satisfies the hypothesis (H), and the

domain of its infinitesimal generator B contains 2)(/4) For x C 3 ( ^ ) we have

Bx = mAx -\- J°° [exp ( ~ ω 0 ξ) T{ξ)x - x] dψ + ax.

Let 0Ce(ί, •) be defined as above by means of the same m and α as £X(ί, )ι

but with φ€{ξ) = ψ(e) for ξ < e and φ£ (ξ) = ιfj(ξ) for ξ > e. Then, by

Theorem 3.3, 0C€(ί, •) is likewise a semigroup in 6 ( ω ) satisfying the conditions

(25). In the notation of §3 we have

llOeU ) I L - | | j8 e (t, )|fc = exp(αί) | | γe (t, . ) 11- ,

where ω{ξ) = ω(f) - ωof, and

Γ 1 logΦ[λ, y β (ί, •)] = »»λ + / e ° ° (expUa - l)dφ.

Arguing as in Theorem 3.1, we see that

Γι log (11 ye ( ί, . ) | f c ) < ω(m) + ( j ^ 1 (Aff - 1) dψ +

+ f~ [exp ΰ(ξ) - l)dψ.

Hence for some K > 0 we have | | θ t € ( ί , ) I U £ e x p ( X ί ) independent of e > 0.

Now φ€ C S ( ω ) . Hence

yβ'(ί, •) B exp(-ί

r,τι*

n=0

is continuous with respect to t in the S(ω) topology, and a fortiori in the strong

topology over Q (ω). Since the shift semigroup1 emt is likewise strongly con-

tinuous, the result holds also for their product γ£ (tf .) = y^(ί, .) * emf We

next set

S * ( O * = fo°° T(ξ)xd0Le(t, ξ) = exp(αί) j °° exp ( - ωoξ) T( ξ )x dγ£(t9 ξ),

so that Se (t)x i s given equivalently by

x T h e s h i f t s e m i g r o u p i s d e f i n e d b y ( ^ β O T t / ) ( ^ ) = / ( ^ ~ ^ ί ) ί h e r e e m t ( ξ ) - 0 f o r< mt a n d = 1 f or f > mt.

364 R. s. PHILLIPS

exp(at) f " exj>{-ωoξ)T{ξ)detm{ξ) fQ°° exp ( - ωoξ) T(ξ)xdγ^ t, ξ)

= exp[(α - ωom)t] T(mt)S^ {t)x,

where

;{t)x = J Γ e x p ( - ωoξ) T(ξ)xdγ'e ( t ,

Since γ^ (ί, £) is uniformly continuous in S ( ω ) , S^(t) is likewise uniformly

continuous, and hence dS£(t)/dt exists in the uniform topology. Thus

B'€ x = / o ~ [exp ( - ωo<f) T( ξ)x - x]dφ€(ξ)

and S ( Z ? g ) = 3£ Since S€{t) is the product of two commutative semigroups

both strongly convergent to the identity as t —» 0, the same is true of S€ ( ί ) ;

and further, for x G 2 > U ) ,

Thus S ( S € ) 3 S)(i4), which is dense in X. Finally, for * G S ( i l ) ,

exp( - ωot)T(t)x- x = exp ( - ω 0^) / T{ξ)Axdξ + ( e x p ( - ωot) - 1 ) * ,

and hence | | exp ( - ωot) T(t)x - x\\ = 0 ( 0 as t —>0. It follows that

β€'* = / e ~ [exp(~ ωo<f) Π £ ) * - * ] <ty

f ~ ~ x ] d ψ ^ B'x,

and that

B€x —>Bx = mAx + J [exp( - ωoξ) T(ξ)x - x]dψ .

F i n a l l y , | | S € ( ί ) | | < II0Ce ( ί , ) |\ω < e x p {Kt). I t n o w f o l l o w s from T h e o r e m

4 . 2 t h a t t h e r e e x i s t s a s e m i g r o u p of t r a n s f o r m a t i o n s U ( t ) s a t i s f y i n g ( H ) a n d

w i t h i n f i n i t e s i m a l g e n e r a t o r B s u c h t h a t S e ( t ) x — > V ( t ) x for a l l x C X . I t

r e m a i n s t o s h o w t h a t U ( t ) = S ( t ) .

L e t u s a p p l y t h e r e s u l t o b t a i n e d t h u s f a r t o 36 = Q ( ω ) a n d { T ( t ) f ) ( u ) =

f ( u — ί ) . T h e s e m i g r o u p T ( t ) s a t i s f i e s ( H ) . N o w w e k n o w t h a t t h e i n t e g r a l

fo°°T(ξ)fda€(t, f ) = fo

Uf(u-ξ)dae(t, ξ)

converges in X to U(t)f. On the other hand, it follows from Lemma 2.2 that

limg^o 0C€(ί, ξ) = CC(ί, £) for each point of continuity of Oί(ί, £). Thus if

/(M) is continuous and differs from zero only on a finite interval then

fo

uf(u - ξ)dθLe(t, ξ) -^jf"/(« - ξ)da{t, ξ)

pointwise and hence in norm. Since this is true on a dense subset in Q(ω), and

since ||θCe(ί, ) | | £ exp(Kί), it follows that Aa€(t, •) converges strongly toAa(t, •) Finally, if we apply Lemma 4.1 once again with

x*[T{ξ)x]CZ(ω)*,

we obtain

x*[Se(t)x]^>f~x*[T(ξ)x]d0L(t,ξ).χ*[S(t)x].

It follows that S(t) = ί/(ί). This concludes the proof of Theorem 4.3.

As a corollary we obtain a partial converse to Theorem 2.3.

COROLLARY 4.1. // α(ί, •) is a semigroup of elements in G(ω) satisfying

(40), then Aa(tf. ) converges strongly to the identity over Q(ω) as t —»0+.

I n t e r m s of t h e p r e v i o u s t h e o r e m , w e s e t 3£ = Q ( ω ) a n d ( T ( t ) f ) ( u ) =

/ ( u - t ) , a r i g h t t r a n s l a t i o n i n Q ( ω ) . T h e n

S(t)f= ζ°T{ξ)fda(t,ξ)= fo

Uf{u-ξ)d*(t,ξ) = Aa(tt.)f.

Hence by Theorem 4.3, ^ α ( ί , •) converges strongly to the identity as t —»0+.

We conclude this section with a discussion of the spectrum of the infini-

tesimal generator B of the semigroup S ( t). We shall need the following lemma.

366 R. S. PHILLIPS

LEMMA 4.2. Let (X(ί, •) be a semigroup in 6 ( ω ) such that the operators

Aa(t9 •) o n C ( ω ) satisfy the hypothesis (H) and let ω ' ( 0 = log | |θ ί( ί , ) | | .

Suppose further that G(ξ) C Q ( ω ) * satisfies the condition (38). 77&erc /or

g(O €1 Q (ω') we

(41) j Γ g ( t ) [ j Γ G ( f ) % α ( t , £)] ώ = /o°° C(f)rf f ( j Γ " g ( t ) α ( t ,

converges in the strong topology over Q ( ω ) .

For / C Q ( ω ) , 0C(ί, ) * / i s continuous for ί > 0 in the Q ( ω ) topology.

Hence the integral of g(t) times a bounded l inear functional on CC(ί, • ) * / i s

equal to the functional on ( f°° g(t)(X(t, )dt)*f; that i s ,

= jΠg(t) [f"G(ξ) (f~f(ξ-u)dua(t, uήdξjdt.

Fubini's theorem permits the interchange of the u and ξ integrations, so that

(42) jΓ [fJ°G(ξ)f(ξ-u)dξ] du(ζa

g{t)a(t,u)dt)

" Γg{t) [Γ (Γ G^f(ξ-u)dξ)dua(t, «)] it.

Again we set / g (ξ) = δ " 1 for 0 £ ξ £ δ and = 0 elsewhere. Then, as in Lemma

4.1,

| G ( f ) / 8 ( f - u ) ι / ί - G ( u ) | < e x p [ ω ( u ) ] e ( δ ) ,

where e ( δ ) — » 0 with δ. Hence, taking the limit in (42) as δ —»0, we obtain

(41).

THEOREM 4.4. Given the semigroup of operators T(t) satisfying ( H ) with

infinitesimal generator A, let Σ ( ^ 4 ) be the spectrum of A and letX(B) be the

spectrum of the infinitesimal generator B of the semigroup S ( ί ) = Θ[θC(ί, • ) ] •

If the Laplace transform of α ( ί , •) satisfies (40), then

where

Φi(λ) = mλ + J ^ ° ( e x p [ ( λ - ωo)ξ] - l)dφ(ξ) + a.

In the proof of this theorem we make use of material developed in an earlier

paper [9]. It was there shown that if λ0 £ Σ(/l) and α C 6(ω), then

Now let R(B; \) be the resolvent of β, and set ω ' ( 0 = log. ||θC(ί, ) | | Then

for R ( λ ) > ωo = inf ω'(t)/t, we have [4, Theorem 1L6.1]

R(B, λ) = ζ° exp(-λt)S{t)dt.

Hence for Λ C Ϊ and x* C 3G* we obtain

x*[R{B ,λ)x] = jΓ°°exp(-λ<) [f"x*[T{ξ)x]dξa(t, ξ)]dt.

We now apply Lemma 4.2 with

G(ξ) = x*[T(ξ)x] C Q ( ω ) * and g{t) = exp(- λt) C δ (ω')

The right side of the above equation can then be written as

fo°°x*[T(ξ)x]dξ

where

J ° ° exp(~λί)0C(ί, •) c?ί Ξ / ° ° e x p ( - λ ί ) ^ α ( ί > .)dt

converges in the strong operator topology in S(ω) over Q(ω). Since this holds

for all x C X and x* £ 3E*, we have

i?(β;λ)= fo°°T(ξ)dξ [ jΓ~exp(-λt)α(t, .)

It follows, for λ0 C Σ (A ) and R (λ) > ωό, that

368 R. S. PHILLIPS

J ^ e x p U o f ) ^ [ j Γ ° e x p ( - λ ί ) α ( ί , •)<

Now if μ = C ( ω ) * , it follows from general integration theory that

e x p ( - λ ί ) C ί ( ί , ) Λ J * / } = SO°° e x p ( - λ ί ) μ[θi{t9-)*f]dt

for all / C C ( ω ) . In particular, let μ(/) = /°° exp (λ o £) f(ξ)dξ, where

R(λ 0 ) < 6^; thenμ(α*/) = Φ(λ, α) μ(/). Hence,°

exp(- λ ί ) α ( ί , )

= j^°° exp(~λθΦ[λ 0 , α(ί, •)] μ(f)dt.

For some / C C(ω) we have μ(f) Φ 0, so that

= j Γ β O e x p ( - λ 0 Φ ( λ 0 , α ( ί f . ) ) Λ - U - Φ ^ λ o ) ] " 1 .

We conclude that

[ λ - Φ 1 ( λ 0 ) ] " 1 C Σ [ / ? ( β , λ ) ] .

It now follows from Theorem 3.1 of [9] that Φί ( λ 0 ) G Σ(J5).

R E F E R E N C E S

1. S. Banach, Theorie des operations line'aires, Warsaw, 1932.

2. S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci.U.S.A. 35(1949), 368-370.

3. N. Dunford and I. E. Segal, Semi-groups of operators and the Weierstrass theorem,Bull. Amer. Math. Soc. 52(1946), 911-914.

4. Einar Hille, Functional analysis and semi-groups, Amer. Math. Soc. ColloquiumPubl. 31, New York, 1948.

5. , Lie theory of semi-groups of linear transformations, Bull. Amer. Math.Soc. 56(1950), 89-114.

6. A. Khintchine, Deduction nouvelle d*une formule de M. Paul Levy, Bull. Univ.d'etat a Moscou. Se'rie Int., Sect. A. 1(1937), 1-5.


7. A. Kolmogoroff, Sulla forma generate di un processo stocastico omogeneo, Atti.Accad. Naz. Lincei, series 6, 15(1932), 805-808, 866-869.

8. Paul Levy, Ύheoτie de I'addition des variables aleatoires, Mo no graph ies desprobabilitie's, fasc. I, Paris, 1937.

9. R. S. Phillips, Spectral theory for semi'groups of linear transformations, Trans.

Amer. Math. Soc. 71(1951), 393-415.

10. N. P. Romanoff, On one parameter groups of linear transformations I, Ann. ofMath., Series 2, 48(1947), 216-233.

11. Kόsaku Yosida, An operator-theoretical treatment of temporally homogeneous

Markoff processes, J. Math. Soc. Japan, 1(1949), 244-253.

THE UNIVERSITY OF SOUTHERN CALIFORNIA

Los ANGELES, CALIFORNIA

UNIFORMLY ACCESSIBLE JORDAN CURVES THROUGH

LARGE SETS OF RELATIVE HARMONIC MEASURE ZERO

GEORGE PIRANIAN

1. Introduction. Let R be a Jordan region in the w-plane, E a set of points

on the boundary C of R, and w - f{ z) a continuous schlicht mapping of the

closed unit disc in the z-plane upon the closure of R, conformal in the open

unit disc. The set E is said to have harmonic measure zero relative to R pro-

vided its image / ι (E ) on the unit circle is a set of Lebesgue measure zero.

Lohwater and Seidel [4] have constructed a Jordan region R whose boundary

passes through a linear set E of positive measure in such a way that E has

harmonic measure zero relative to R. Also, Lohwater and Piranian [3] have

described a Jordan region R whose boundary passes through a set £ of positive

two-dimensional Lebesgue measure in such a way that E has harmonic measure

zero relative to R. In both cases, the set E consists of points each of which is

not the end point of any rectifiable arc whose remaining points lie in R. And

in both cases the fact that the points of E are not accessible by rectifiable

paths in R serves no other purpose than to permit the application of Lavrentiev's

theorem on finite accessibility [l] (see also Tsuji [9, p* 99] and Seidel and Walsh

[8, p. 143]) in proving that the set E has harmonic measure zero relative to R

The author of the present note holds the view that bizarre examples are

acceptable only as long as no simple replacements are available; and he be-

lieves that the geometrically important aspect of the earlier constructions is not

the fact that the points of E are not finitely accessible from /?, but merely this,

that the points of E cannot be reached from the interior of R except along paths

through narrow corridors (whose tortuous character is irrelevant, except in the

proofs). The motivation of this belief is evident from the proof that will be given

of the following proposition.

THEOREM 1. There exists a Jordan region R whose boundary C passes

through a linear set of positive Lebesgue measure and of harmonic measure zero

relative to R, and which has the property that each point of C can be reached

from a certain fixed point in R by an arc whose length is less than one and

whose interior points all lie in R.


371

3 7 2 GEORGE PIRANIAN

In §2, this theorem is proved by means of the principle of the maximum. In

§3, a stronger theorem is proved with the aid of Lavrentiev's theorem on angular

accessibility. The last section proves a theorem concerning Jordan curves pass-

ing through sets of positive two-dimensional measure, and presents a conjecture

suggested by the proof of the theorem.

2. Proof of Theorem 1. Let L be the line segment joining the points w =

± 1/2 + i, and let P u denote its midpoint. Let an open segment of length 1/4

and with its midpoint at P u be deleted from L, and let P 2 1

a n (^ 22 be ^ e m ^ "

points of the two surviving segments, with P21 to the left of P 2 2 . Let open seg-

ments of lengths 1/16 and with their midpoints at P 2 1

a n ^ 22 be deleted from

what remains of L, and let P3r (r = 1, 2, 3, 4) denote the midpoints of the sur-

viving segments, with P3r to the left of P3,Γ+i Let segments of lengths 1/64

and with their midpoints at P 3 Γ (r = 1, 2, 3, 4) be deleted from what remains of

L. Let this process of deletion be continued indefinitely, and let E denote the

Cantor set on L which survives the process. Clearly E has Lebesgue measure

1/2.

For n = 1, 2, , and r = 1, 2, , 2"" 1, let pnτ be the projection of the

point Pnr upon the line v = 1 - 2 " Λ , where v denotes the imaginary part of w.

Let /?! consist of a (closed) narrow rectangle Qn in the w-plane, with its sides

parallel to the coordinate axes, and with its short sides passing through the

points w = 0 and w = p u = i/2, respectively. Let R2 consist of Rι together

with two disjoint narrow parallelograms Q2l and (λg, where Q2ί has one of its

short sides on the upper side of Qn while its other short side passes through

p 2 1 ; and where Q22 has one of its short sides on the upper side of Qn while its

other short side passes through p 2 2 (that is, let R2 be in the shape of the letter

Y, with its base at the origin, its fork at p u , and its upper ends at p 2 1 and P22).

Let R3 consist of R2 together with four narrow parallelograms Q3r (r = 1, 2, 3, 4)

of which the first two reach from the free end of Q2i to p 3 1 and p 3 2 , respectively,

while the other two reach from the free end of Q22 to p 3 3 and p ^ , respectively.

Let R denote the interior of the set which is obtained as this construction is

continued indefinitely. Then R is a Jordan region in the shape of a tree whose

trunk undergoes dichotomous branching near p u , whose branches split again

near p 2 1 and p 2 2 , and so on. The boundary C of R meets the line-segment L in

the set E. And each point of C can be reached from the point w = 2ι/5 by a

curve in R of length less than 1.

It remains only to make certain that the harmonic measure of the set E rela-

tive to the region R is zero. Now let tυ = f(z) map the unit disc conformally

upon R, with /(0) = i/4. For n = 2, 3, , let Cn denote that part of the

boundary of R which lies above the line v = 1 - 2ι~n. Then, by the principle of

UNIFORMLY ACCESSIBLE JORDAN CURVES 3 7 3

the maximum, the measure of the inverse image on | z \ - 1 of the set Cn can be

made arbitrarily near to zero simply by choosing the parallelograms Qnτ suf-

ficiently narrow (see [6], p. 63). This completes the proof of the theorem.

3. On star-shaped Jordan regions. The Jordan region constructed by Lohwater

and Seidel has boundary points that are not accessible from the interior by recti-

fiable arcs. The boundary points of the region constructed in the preceding

section are not only finitely accessible from the interior, but they are uniformly

finitely accessible. However, the new region still leaves something to be de-

sired, because not all of its boundary points are linearly accessible from the

interior.

THEOREM 2. There exists a star-shaped Jordan region R whose boundary

meets the circle \w\ = 1 in a set of positive Lebesgue measure and of harmonic

measure zero relative to R.

Let J?o denote the disc \w \ < 1, and let Ri be obtained from Ro by the de-

letion of an isosceles "triangle" whose boundary consists of an arc of the

circle | w \ - 1 and of two rectilinear segments of length 1/2. Let R2 be obtained

from /?! by the deletion of finitely many further such isosceles triangles with

one side on the circle | w | = 1 and with rectilinear sides of length 1/4. Let R

be obtained by the continuation of this construction, and let appropriate pre-

cautions be taken so that the boundary of R meets the circle \w\ - 1 in a set

E of positive Lebesgue measure, and so that no point of E is the vertex of an

ordinary triangle whose interior points all lie in R. Because R is star-shaped,

it follows from Lavrentiev's theorem on angular accessibility [2, see part 1 of

Theorem 1 on p. 822] that E has harmonic measure zero relative to /?, and the

proof is complete.

The construction that has just been carried out was suggested by a region

of Lusin [5]. However, the latter region is not star-shaped, as the reader will

discover if he passes from the examination of Lusin's figure on p. 153 to the per-

usal of the accompanying text.

4. The two-dimensional analogue of Theorem 1. The method used in proving

Theorem 1 can be used to extablish the following proposition.

THEOREM 3. There exists a Jordan region R whose boundary C passes

through a set of positive two-dimensional Lebesgue measure and of harmonic

measure zero relative to /?, and which has the property that each point of C can

be reached from a certain fixed point in R by an arc whose length is less than

one and whose interior points all lie in R.

3 7 4 GEORGE PIRANIAN

Let Ro denote the open square with vertices w = 0, 1/2, ι/2, and (1 + i)/2.

Let /?! denote the set of points x + iy in Ro which satisfy one of the two condi-

tions 0 < x < p9 0 < y < p , where p is some small positive constant (p = 1/10,

for example). The region R shall consist of /? t together with a certain set of

narrow strips whose closure nearly fills the remainder of the square RQ. For the

sake of a simple description, the language used by Osgood [7] in his construc-

tion of a Jordan curve of positive area will be applied. The region Rx is to be

thought of as constituting a body of water. A narrow channel R2 from the hori-

zontal leg oί Rί to the upper boundary of Ro bisects the part of Ro not occupied

by Rt. To the left of R2 and immediately adjacent to it a dike Q2 is erected, of

equal width and length with R2. The portion of Ro which has not yet been used

consists now of two rectangles. One of these is bisected by a horizontal channel

R3 extending from Rχ as far as Q2; the other is bisected by a horizontal channel

R4 extending from R2 to the right-hand edge of Ro. Parallel to each of these two

channels, and immediately adjacent to it from below, a dike is erected.

The portion of Ro which has not yet been used consists now of four rec-

tangles; each of these rectangles is bordered by channels on the left and below,

and by dikes (or edges of Ro) on the right and above. The cooperative reader

recognizes the spirit in which the construction of channels and dikes is to be

continued. The union R of all the channels constructed in the process consti-

tutes a Jordan region. All points lying on dikes are exterior to /?, and all the

"swampy" points of J?o which lie neither in channels nor on dikes are boundary

points of/?. If at each stage of the construction the new channels and dikes are

made narrow enough, the boundary C of R passes through a set of positive two-

dimensional Lebesgue measure.

Also, the principle of the maximum implies that if the channels are suf-

ficiently narrow, then under every conformal mapping w - f(z) of the unit disc

upon the Jordan region /?, a set of measure 2π on \z \ = 1 is mapped into a set

E on C which lies on a certain denumerable collection of rectilinear segments,

that is, which has two-dimensional measure zero. It follows that a set of meas-

ure zero on | z | = 1 is mapped upon a set of positive two-dimensional measure.

Since C is obviously uniformly accessible from the interior of /?, the proof is

complete.

In the case of the Jordan region R constructed for the proof of Theorem 3,

a set of two-dimensional measure zero on the boundary C is the image of a set of

one-dimensional measure 2π on the circle \z\ = 1. This state of affairs gen-

erates a disquieting suspicion: it may be that Theorem 3 is in a certain sense

a trivial consequence of the fact that some Jordan curves have positive two-

dimensional measure zero. Of many modes of misbehavior of functions on the

UNIFORMLY ACCESSIBLE JORDAN CURVES 3 7 5

boundaries of their domains of existence, it is true that all of the misbehavior

necessarily takes place exclusively on a small set. In the light of what is al-

ready known, the following proposition appears to be reasonable:

C O N J E C T U R E . Let

oo oo

f ( z ) = Σ 0 * * * > w^th Σ n \ a n \ 2 < oo,n =o n - o

and let E be the set of points e i θ for which

f{eiθ) = lim f(reiθ)

exists. Then E has a subset E* of measure 2π and with the property that the set

of values { f ( e ) } {eι on E*) has two-dimensional Lebesgue measure zero.

REFERENCES

1. M Lavrentiev, Sur la correspondance entre les frontieres dans la representationconforme, Mat. Sbornik 36(1929), 112-115.

2. , Sur quelques proprietes des functions univalentes, Mat. Sbornik N. S.1(1936), 815-846.

3. A. J. L oh water and G. Piranian, Con formal mapping of a Jordan region whoseboundary has positive two-dimensional measure, Michigan Math. J. 1(1952), 1-4.

4. A. J. Lohwater and W. Seidel, An example in conformal mapping, Duke Math. J. 15(1948), 137-143.

5. N. Lusin, Sur une propriete des functions a carre sommable, Bull. Calcutta Math.Soc. 20(1928), 139-154.

6. R. Nevanlinna, Eindeutige analytische Funktionen, Berlin, 1936.

7. W. F. Osgood, A Jordan curve of positive area, Trans. Amer. Math. Soc. 4(1903),107-112.

8. W. Seidel and J. L. Walsh, On the derivatives of functions analytic in the unitcircle and their radii of univalence and of p valence, Trans. Amer. Math. Soc. 52(1942)128-216.

9. M. Tsuji, On the theorems of Caratheodory and Lindelof in the theory of conformalrepresentation, Jap. J. Math. 7(1930), 91-99.

T H E UNIVERSITY OF MICHIGAN

NOTE ON SOME TAUBERIAN THEOREMS OF 0. SZASZ

C. T. RAJAGOPAL

1. Introduction. The object of this note is to record extensions of the

Tauberian theorems for Abel summability which form the subject of a recent

paper of Sza'sz in this journal [6, Theorem 2] The extensions given as Theorems

II, III', concern a process of summability which may be called the (Φ, λ)—proc-

ess, discussed elsewhere [3] and defined below. Theorems II, III' include simi-

lar results, given as Theorems I, III, which are implicit in [3]

The process of (Φ, λ) — summability is defined for any real series 2* On as

follows.

Let φ(u) satisfy the following conditions:

C ( i ) φ{u) is positive, continuous and monotonic decreasing in (0,oc);

C(i i) φ(0) = 1, J { φ{u )/u! du is convergent for every € > 0

C(ii i) φ(u) has a continuous derivative -φ{u) in (0,00), this derivative

being, on account of ( i), negative and such that

φ(u) = f~ φ(x)dx;

C(iv) φ(u) is monotonic decreasing and has a continuous derivative in

(0,oo);

C(v) J°° uix φ(u)du £ 0 ( - c o < x < 0 0 ) .

L e t

Φ λ ( O = Σ an φ ( λ n t ) , t > 0 , 0 = λ 0 < \χ < λ 2 < ••• , λ n — > 0C .

Then Z*n = ι an is said to be (Φ, λ)~sumrnable if

lim Φ-χ(O exists.

Received January 22, 1952.Pacific J. Math. 2(1952), 377-384

377

378 C. T. RAJAGOPAL

Examples of φ(u) which satisfy the conditions (C) are furnished by

φ(u) = e~u [Abel-Laplace kernel],

φ(u) = (1 + u)~p, p > 0 [Stieltjes kerenel],

φ(u) = u/(eu - 1), u > 0, φ(0) = 1 [Lambert kernel].

The theorems of this note rest primarily on a result proved in my note already

referred to [3, Theorem 2] and running as follows:

THEOREM A. Let φ(u) fulfil the conditions C ( i ) - ( v ) . Suppose that A {u)

is a function of bounded variation in every finite interval of ( 0, oo), A ( 0 ) = 0. //

( 1 ) lim inf - / " u d\A(u)\ > - oo,U —too U 0

and

(2) Φ ( ί ) = f°° φ(ut) d{A (u)\ exists for t > 0 and converges as t—» + 0 ,

then we have

Λx(u)( 3 ) lim = lim Φ ( ί ) , where Ax(u) ~ f A(x)dx.

U-*oo U £->+0 ^0

2. Extensions of Szsfez's Tauberian theorems. It is clear that Theorem A

can be restated for a λ-step function defined thus:

A (u) = ax + α2 + . . + an for λn < u < λ Λ + 1 ,

(4)

A(u) = 0 for 0 s a s x —> °°»x κ<_*

Aγ(x) being alternatively defined as in (3), and if, as n —»oo, we have

NOTE ON SOME TAUBERIAN THEOREMS OF O. SZA'SZ 3 7 9

n

Σ ( K l - « v ) p AP ( λ v - xv-t)l~p = 0 ( λ B ) (P > l ) ,

1

(5)

π=i an converges to s.

If, in Theorem A, A (u) is the step function of (4), then (1) is a consequence

of

( I 7 ) lim inf Y α v > - oo,

which in turn is a consequence of the first condition in (5), as shown by Szasz

[5, p. 126], while (2) becomes the hypothesis of (Φ, λ) — summability of 2* an.

Thus Theorem A, in conjunction with Lemma 1, yields the following result.

THEOREM I. // Σ T = i a" ι s ^ ' ^^"" s u m m a ^ e t o s? then (5) implies that

ΣT=i ajι i S converSenί; to s

Theorem I was proved by Szasz in the case φ(u) = e~u [5, Theorem 4 ( a ) ]

It is the case λn - n, φ(u)~ e~u of this theorem which he has recently gener-

al ized [6, Theorem 2]. Repeating his arguments, in a sl ightly more general form,

we obtain the following extension of Theorem L

THEOREM Π. // Σ Λ = I an is (Φ> λ) — summable to s and if, as n —-»oo, we

have

( 6 ) Un = ^ λ v (\av\ - av)

' ' ' T — "" ••— —> 0 whenever "•— —> 1 >^ λ

ί/ien 2*n=ι an ι s convergent to s.

Theorem II is an extension of Theorem I in the sense that the hypotheses of

the former cover those of the latter. This observation, contained in the next

lemma, can be substantiated exactly as in its particular case for which λn = n

380 C. T. RAJAGOPAL

[6, p. 119].

LEMMA 2. The first relation in (5) implies the first relation in (6), while

(5) as a whole implies (7) through the relation

um - un xm—> 0 as >1 {n —>oo).

λn λn

For the proof of Theorem II we require a further lemma which is virtually

contained in Sza'sz's proof of the case λn - n, φ(u) = e~u of Theorem II.

LEMMA 3. // Σrc=i an i S (^» K 1)"" summable to s9 and the second con-

dition of (6) holds along with (7), then Σrc = i an ι s convergent to s.

Proof. The hypothesis of (/?, λ, 1) — summability of 2* an to s implies that

1 n nσn = ^ ( λ v + 1 - λv) sv — > s , where sn = £ α v .

Now, we have the identities

Σ (λπ+λ + i ~" λn+v) an+v ,

Kσn - σn-k-ι)

1+

By using these identities, exactly as they have been used by Szasz in the case

λπ = Λ L6, pp. 118-119, pp. 120- 121], we can prove that

lim sup sn _< s, lim inf sn >_ s ,n—>oQ n —»oo

whence the conclusion of Lemma 3 follows at once.

Proof of Theorem II. From the first condition in (6) we get

n n

- Σ λ v α v < £ λ v ( | α v | - av) ^ Un = O ( λ n ) .

Therefore, defining A (u) in Theorem A as the λ-step function of (4), we observe

that hypothesis (1) of the theorem obtains as a result of ( l ' ) Since hypothesis

(2) also holds, in the form of the (Φ, λ) —summability of Z* ««> we are led to

conclusion (3) in the form of the (/?, λ, 1) — summability of 2* a>n The desired

conclusion now follows from Lemma 3.

3. A second generalization of Theorem I. There is a generalization of

Theorem I, different from Theorem II and very similar to a theorem of Delange

[l , The'oreme 8] which may be regarded as one more result of employing the

technique of Sza'sz embodied in Theorem A and in certain arguments, already

cited, involving (/?, λ, 1 )-summability [5, pp. 126-127].

THEOREM I ' . // ΣrΓ=i an ι s a series (Φ, X)summable to s, and if

n

( 5 ' ) Σ ( U v l - α v )P λP ( λ v - λ v - x ) 1 ^ = 0 ( λ n ) , p > l , n—»oo,1

then \sn\, the sequence of partial sums of ^ Λ = i an > ι s such that

lim sup sn = 5, lim inf sn = s — I

where

I - lim sup (I an \ - an ) / 2 .n->oo

The proof of Theorem I ' is like that of Theorem I, but uses (in conjunction

with Theorem A) the following lemma instead of Lemma l

LEMMA l ' // Σ/ ι=i an ι s summable (R, λ, 1) to s and satisfies (5'), then

the sequence \sn \ of its partial sums behaves as in (8).

Proof. The first half of Szasz's arguments [5, p 126] proving Lemma 1, with-

out any modification, establishes the first conclusion of (8).

To obtain the second conclusion of (8), we note that / >_ 0 by definition and

/ < oo by (5') as shown in Lemma 4 which follows this proof. From the fact

382 C. T. RAJAGOPAL

\<*n\ - an

we then infer that

lim inf an >_ — Z.Λ-»oo

We thereafter employ the second half of the aforesaid arguments of Sza'sz [5,

pp. 126-127] and reach the conclusion

(9) lim inf sn > s - Z.

When I = 0, (9) and the universal relation, lim inf sn _< s, together establish the

second conclusion of (8). When Z > 0, we see that there is an increasing se-

quence of positive integers nί9 n2, ••• , nn c such that, as r—> oo, | anf \ -

anγ—> 2Z which implies (since Z > 0) that aUf is ultimately negative and aUr —>

- Z. Hence, when Z > 0,

(10) lim inf sn < lim sup sUr = lim sup (sτlr-.ι + aUr)

= lim sup sn -! - I £ s - I.Γ-*oo

Consequently, in the case Z > 0, the second conclusion of (8) follows from (9)

and (10). The proof of Lemma 1' is now complete.

Theorem I is a special case of Theorem V with Z = 0 as we can see from the

following plain statement.

LEMMA 4. // Σ n = i an is anJ (real) series satisfying (5'), then

In particular, (5') and the condition lim λn/λn~ι - 1 together imply that

I = lim sup (t an \ - an ) /2 = 0.

Now, in any (real) series ^ an,

- a

nmin (0, an)

so that lim inf an >_ 0 implies / = 0 and conversely. Thus, from Theorem I ', we

can say that there is a variant of Theorem I with the second condition of (5)

replaced by lim inf an >_ 0; in fact, we can say more as follows.

For series Σn=ι an summable (Φ, λ) and satisfying the Tauberian condition

(5'), a necessary and sufficient convergence condition is lim inf an >_ 0.

In the above statement, (5') can be replaced by the following simpler con-

dition which implies (5')*.

an λn

lim inf > — oon-*oo \ n - \ n ^ ί

4. A generalization of Theorem II. Following Szasz, we have seen that

conditions (6) and (7) of Theorem II together include the corresponding condition

(5) of Theorem I. Following Szasz further [6, §§4.5], we see that conditions

(6) and (7) together can be expressed in the Schmidt form, and the following

equivalent of Theorem II obtained.

THEOREM IΠ. // Σn=ι an ι s (Φ> λ)-summable to s and if, as n —>oo,

(11) —>oo, jΓ ( | α ^ | - a>v)—*0 wheneverλ

then 2*n=ι an ι s convergent to sum s.

A generalization of Theorem III, related to it as Theorem I ' is to Theorem I,

may now be stated in a familiar form as under.

THEOREM IIΓ. // Σ Γ = i an ^s a series (Φ, Xysummable to s and such that

either lim lim sup 2* ( | α v l ~ " α v ) = 0 ,

8+< λ v<\n<(l+S)λn

(12)

or lim lim inf min ^ α^ > 0,S ^ + o n^oo λre + l l λ v < λ m < ( i + δ ) λ n

then the sequence { sn } of partial sums of the series satisfies (8), provided that

I s lim sup (I an \ - an ) / 2 < oo.n —»oo

Since — av _< \av\ — α v , t h e first alternative of (12) implies the second, and

the second is the only alternative that need be considered. Now the second

3 8 4 C. T. RAJAGOPAL

alternative clearly implies that, on the assumption λ n + ι / λ n — > 1, we have

lim inf an >_ 0 and hence I = 0. Therefore, in Theorem III ' , we can drop the

explicit assumption / < oo and assume instead, either

lim λn + ι/λn = 1, or lim inf an >^ 0,

getting the two cases of I = 0 in the following corollary of which the first case

is Theorem III.

COROLLARY III ' . // a series 2*n=ι an is (Φ> λ)-summable and such that

(12) holds along with

either \imλn^.ι/λn - 1, or lim inf an >_ 0,

then 2*n=ι an ι s convergent to s.

Corollary III ' , in the case φ(u) — e u, is a classic theorem of Szasz [4,

p. 338]. Theorem III' in the same case and in an incomplete form has been al-

ready obtained by me [2, Theorem F]; its proof readily suggests the following

proof of Theorem ΠI' in its generality.

Proof of Theorem HI ' . Confining ourselves, as we may, to the second al-

ternative of (12), we can show that this alternative and the condition I < oo

(equivalent to lim inf an > — oo) together imply ( I 7 ) , using an argument in-

dicated elsewhere [2, Lemma 1] After this we can appeal to Theorem A. and

infer that 2^ an is (R> λ, l)-summable to s. The subsequent completion of the

proof is along the lines of the proof of Lemma 1 ' .

REFERENCES

1. H. Delange, Sur les theoremes inverses des procedes de sommation des seriesdivergentes {Deuxieme Me'moire), Ann. Sci. Ecole Norm. Sup. (3), 67(1950), 199-242.

2. C. T. Rajagopal, On some extensions of Ananda Rau*s converse of Abel's theorem,J. London Math. Soc, 23(1948), 38-44.

3. ,A note on generalized Tauberian theorems, Proc. Amer. Math. Soc, 2(1951), 335-349.

4. O. Sza'sz, Verallgemeinerung und newer Beweis einiger Sdtze Tauberscher Art,Sitzungsber. Bayer. Akad. Wiss., Naturw. Abt. Mϋnchen, 59(1929), 325-340.

5. — , Converse theorems of summability for Dirichlet series, Trans. Amer.Math. Soc, 39(1936), 117-130.

., On a Tauberian theorem for Abel summability, Pacific J. Math., 1(1951),

RAMANUJAN INSTITUTE OF MATHEMATICS

MADRAS, INDIA

A MODIFICATION OF THE NEUMANN-POINCARE METHODFOR MULTIPLY CONNECTED REGIONS

H. L. ROYDEN

1. Introduction. Some interest has been attached to the problem of effective

computation of the solution to boundary value and conformal mapping problems.

Birkhoff [3; 4] has given an excellent iteration precedure for the solution of the

conformal mapping problem for simply connected regions. However, the con-

vergence of his method is easily seen to be the same as that for the classical

Neumann-Poincare method in potential theory, which, while converging for all

simply connected regions [l;5], fails to converge for the computation of the

harmonic measures of multiply connected regions [l;2]. Since it is primarily

these harmonic measures which are needed in the conformal mapping problem, we

derive in the present paper a modification of the Neumann-Poincare method which

will converge in this case and apply it to the conformal mapping of doubly con-

nected regions. While the formulas involve certain ^-series, these should not

present a major problem for numerical computation since the series are very

rapidly convergent and only a few terms need be taken.

2. General formulation. Let Ω be a multiply connected region which is

bounded by a smooth curve C. Let U (z) be a continuous real-valued function

defined for z £ C. We wish to consider the problem of finding a function u + iv

which is analytic in Ω and for which

(1) u(ζ)—>U(z), as ζ^zCC.

We do not require that v be single valued in Ω, and we shall assume that at a

fixed point ζ0 £ C we have

(2) l/(£0) = 0.

L e t IF be a Riemann surface which contains Ω and for which Ω = W — Ω is

also a connected region. For example, if Ω is the plane region exterior to the

contours Cl9 ••• , Cn-t and interior to Cn, we may take W to be the double of

the region R which is bounded by circles γl9 •• , y r ι , where γι l ies inside

Received September 12, 1951. Work done under contract N5ori-07634, between theOffice of Naval Research and Harvard University.

Pacific J. Math 2 (1952), 385-394

385

386 H. L. ROYDEN

C, for i = l , , n — 1, and γn lies outside Cn.

If g(z;£, <£Λ) * s t n e Green's function for IF, that is, a harmonic function of theovariable z which has a positive logarithmic pole at ζ and a negative logarithmic

pole at ζ 9 we have the symmetry relation

(3) g(z; ζ, ζ0) - g U 0 ; 6 £ 0) = g(ζ; z, zQ) - g(£0; z, zQ).

From this it follows that the derivatives of g with respect to x and y (where

z = x + ίy) are harmonic functions of ζ

Thus if μ(z) is a continuous real-valued function defined for z £ C, then the

function

(4) α.(() = / μ(z) ds (<Γ£Ω),1 2π J c dnz

z

(where n is the inner normal to Ω) is a harmonic function of ζ. If further we

define

1 r dg(5) u (£) = J μ(z) —— rfs (^CΩ),

e 2ττ c dnz

z

then we know by the well-known boundary behavior of double layer distributions

that

due(ζ) dι(ζ)(6) — — = — — (ζCO,

dn dn

while

(7) , ( 0 . - 1 , ( 0 •!,,(<,) • £

and

If we suppose that μ(£) = 0, then the solution of our problem is determined

if we can solve the integral equation

(9) 2U(ζ) = -μ(ζ) + - f μ(z) A dsz,π L dnz

for μ subject to the condition that μ(ζQ) = 0. If this is done, then u is given by

NEUMANN-POINCARE METHOD FOR MULTIPLY CONNECTED REGIONS 3 8 7

(7), and υ is given by

(10) v = — / μ{z) — ds ,2π L onz

where dh/dnz (z;ζ,ζQ) is the harmonic conjugate with respect to ζ of dg/dnz {z;

ζ, ζQ); that is, such that

_dg_ m dh

dn dn

is an analytic function of ζ.

3. Solution of the integral equation. Because of the smoothness of C, we

know that dg/dnz is a continuous function on C, and the Fredholm theory states

that either

(ID W(ζ) = - μ(ζ) + - / μ(z) p - dsf

π ϋ σnz

has a unique continuous solution for all continuous functions U(z), or else

(Λ->\ ( r\ λ C ( \ d g J\* *) μVζ ) = — J μ\z) as

π c dnz

has a nontrivial solution. Moreover, if we set

(13) T[φ] = — f φp- dsz,77 c o n z

we have

(14) /zU) = - 2 2 λ Γ ^>v = o

uniformly in z for all λ with | λ | < | λ o | , where λ 0 is the absolutely smallest

value of λ for which (12) has a nontrivial solution. Thus if we can show that

I λ 0 I > M > 1, we know that the series

(15) μ ( z ) = - 2 £ Tv U

converges with order at least 1/M and gives a solution to the integral equation

(9). For computational purposes, the series (15) is more conveniently expressed

388 H. L. ROYDEN

by the following iteration:

ι

(16)

μn(ζ) = - n χ

In order to ensure the validity of this iterative method, we need only to bound

the eigenvalues of (12) away from unity. Suppose that we have an eigenfunction

Φ = λ ι TΦ .

Then

and

are harmonic functions in Ω and Ω, respectively. For ζζ2 C we have, by (7) and

(8),

1Uί 2

Also,

whence

(17) «,.(C) = — — -1 + λi

By (6) we have

9 9

and

D(ui) = J ut ds = — f ue ds = — — D(ue),^ on 1 + λi c σn λi + 1

NEUMANN-POINCARE METHOD FOR MULTIPLY CONNECTED REGIONS 389

where

and

Thus

" ' - > -

u Uϊ

dx

2

dy

2

+ dxdy

due

dx dy.

where

ΏUe)

Since D (ue) and D (MJ) are always nonnegative, we conclude that λt is real

and

(19) > 1.

In order for equality to occur in (19), we must have either ue constant or u(

constant. In either case the constant must be zero since ui (ζ ) = 0 and ue (ζQ) =

0, while Ω and Ω are connected. Thus equality is excluded in (19) and we have,

taking λx to be λo» the absolutely smallest eigenvalue,

(20) Kthis ensures the convergence and correctness of the iteration (16).

In order to obtain an estimate for | λ 0 | and consequently for the rate of con-

vergence of (20), we suppose that Ω and Ω are topologically equivalent. Then

following Ahlfors [l] we take a quasi-conformal mapping f(z) of Ω onto Ω with

dilation quotient < K. Then the functions V{ and υe, which are the harmonic

conjugates of ui and ue with the conditions that viiζ ) = ve {ζ ) = 0, are har-

monic functions in Ω and Ω with the same values on C by (6). Moreover, Ό{VJ) =

Dim); D(ve) = D{ue).

But now ve [f(z) ] is a continuous function in Ω which has the same periods

and boundary values as v{. Hence by the Dirichlet principle we have

D{ve[f(z)]\ >D\Vi\.

390 H. L. ROYDEN

But

Hence

DlvΛfiz)] 1 <KD{v\.

K >D{ve

In a similar manner, using f , we obtain

K >D\ve

D{vi

λ 0 -

λ 0 +

λ 0 +

λ 0 -

Hence either

that is,

(21)

- (K + 1) K + 1λ 0 < , or λ n >

0 - Uί + l ) ° - x - l

K + 1

K - 1

4. The doubly connected case. We now suppose that Ω is a doubly connected

region in the plane contained in the circle \z\ < 1 and containing the circle

I z I S. ? 2 within its inner contour. We take our Riemann surface W to be the torus

formed by identifying the points z and q2z.

Now we shall show that the Green's function for W is

1 lθg I z

2 log?log(22) g ( z ; ί , C 0 ) ^ l o g

where

/I \ °° 2(23) tf#(l) = ^ 4 1 — log ί , g) = 1 + Σ ^~ ) Λ 9Λ ( ί Λ + t~n

From (23), we see that ϋ^. is a single-valued function of t and that

+2Π 1 Σ <-

Hence

«(? «; 6 Co) = log2 log

log

so that g is a single-valued function of z on the torus. By (23) we see that g has

poles at ζ and ζ 9 since

Moreover, ^ 4 has only one zero for (1/2 i) log t in the rectangle whose corners

are a9 a + π, a + π - i log q, a - i log q [6, p. 465]. Thus ϋ-Jj) has only one

zero for q2 < \t\ < 1, and consequently g is the Green's function of W.

5. Computation. The iteration (16) becomes now

(24) μ χ { ζ ) = - 2 U { ζ ) ,

- / μ n . ( z ) A — z ' ( s ) ώ ,π L dz

since

vhile

and so

^/i dz dn dz dn dz dn

dz dz= i

dn d s

i ! - 2 Rdn

- 2 « &

dg dz

dz ds

dz

Clearly (24) is independent of the choice of the parameter s, and hence

need not be arc-length, but may be any conveniently chosen parameter σ

Now1 log

(25) ' " "M*/O loglog

392 H. L. ROYDEN

whence

(26)

where

(27)

2 - ^ =/U, ζ) -f(z, ζ0),

f(z, ζ) =-ϋ'^qz/ζ) q/ζ + 1 log I ί I

#tiqz/ζ) 2z log q

Thus (24) becomes

(28) μιU) =

~ i {μn-t(z) Ufiz,ζ)-f(z,ζ0K z'ds

This is equivalent to

(28') - μ[0)-

Near z - ζ the boundary has the following expansion in terms of arc length

s:

(29) O(s3) ] ,where primes denote differentiation with respect to arc-length and K is the curva-

ture at ζ. Then, since

have

C

Also

vhile


Hence

(30)

whence

(31)

/ y *

γ = - j - [ 1 + iKS + 0{s2)].

qz'/ζ 1 C_ i κ s _ i _2 ζ

O(s2)

In terms of an arbitrary parameter σ, this becomes

(31') A * ) z ' = — κ(z) — - il —2 dσ z

2 log q

1 log

2 log

where now the primes denote differentiation with respect to σ.

For computational purposes, it would seem best to calculate

(32)

and

(33)

(->" 1n

= l

= 1

for the different values of ί = qz/ζ which occur. Then

(34)t&'Λt) . 1 log I ζ I

The conjugate function v is given by the integral

(35)

where

(36)

and

v = J μ(z) & f(z, ζ) z' do ,277 C

f(z,ζ) * ' =2 log

3 9 4 H. L. ROYDEN

(37) A / U z) z' ,

σ

with primes denoting differentiation with respect to σ. The integral (35) has to be

taken to mean its Cauchy principle value if ζ C C. However, since

/ A f(z,ζ) z' do = 0 ,

we may write (35) as

(38) v = -ϊ- jΓ [μ{z) - μiζ)} & f{z, ζ) z'dσ,

with

6. An application to conformal mapping. In order to map the doubly connected

region Ω onto the annulus r < | w \ < 1, it is only necessary to find, say using

the method of the preceding section, that analytic function u + iv in Ω for which

u = 0 on the outer boundary of Ω and u = 1 on the inner contour. For if ω is the

period of vf then

(2π/ω) (u +iv)w = e

maps Ω onto the annulus, and v/ω gives the angular correspondence on the

boundaries.

REFERENCES

1. L. Ahlfors, Remarks on the Neumann-Poincare integral equation, this issue p.273-282.

2. S. Bergman and M. Schiffer, Kernel functions and conformal mapping, CompositioMath. 8 (1951), 205-249.

3. G. Birkhoff, D. M. Young and E. H. Zarantonello, Effective conformal transforma-tion of smooth, simply connected domains, Proc. Nat. Acad. Sci., 37 (1951).

4. , Numerical methods in conformal mapping. To appear in NBS AppliedMath. Series 18.

5. H. Poincare* La methode de Neumann et le probleme de Dirichlet. Acta Math. 20(1897), 59-142.

6. E. T. Whittaker and G. N. Watson, Modern analysis, Cambridge, England, 1940.

HARVARD UNIVERSITY

A THIRD ORDER IRREGULAR BOUNDARY VALUE PROBLEM AND

THE ASSOCIATED SERIES

GEORGE SEIFERT

1. Introduction. Certain problems in aeroelastic wing theory [l] give rise

to a third order irregular boundary value problem of the form given in equation

(1) below. Questions have been raised [1] as to conditions under which functions

have an expansion in terms of the associated characteristic functions. It is

shown in this paper that the general approach by L. E. Ward [2] in dealing with

a somewhat more specialized problem can be suitably modified to provide an

answer to these questions.

We are concerned with the differential boundary value problem

(1) L(u(x),λ) = a ' " (*) + p O O a ' U ) + (q(x) + λ)u(x) = 0,

n ( 0 ) = u' (0) = u " (1) = 0,

w h e r e p ( x ) = x ψ ί ( x 3 ) , q ( x ) - φ 2 ( x 3 ) , a n d ψ ι ( z ) a n d φ 2 ( z ) a r e r e a l f o r

real z and analytic on | z \ <_ 1. We seek conditions on f{x) such that it be

expansible in terms of the characteristic functions of (1) and its adjoint.

We shall first need a number of definitions and lemmas. Define:

i) δ 3 (ί) = eωχt t ω^

8t(t) = - S5(O,

where ωγ = - 1, ω2 = eπi/\ ω3 = e~πi/3;

ϊi) Δ ( Λ , ί, p) = p " 1 83[f}(x-t)] r{t) ~ δ 2 [ p U - ί ) ] p ( O

where r(t) = q(t) - p'(t), and the complex number p satisfies

p 3 = λ, |arg p | < τr/3;

Received October 8, 1951. The author wishes to thank Professor L. K. Jackson forhelpful suggestions towards clarification of the notation used.

Pacific J. Math. 2 (1952), 395-406395

396 GEORGE SEIFERT

iii) the regions Sί and S2 of the p-plane by 0 £ arg p £ π/3 a n d - 77/3 £

arg p £ 0, respectively.

We shall be concerned with the integral equation

(2) Γ A(x,t,p)u(t,ξ,p)dt.3p ξ

2. Lemmas. We shall use the following results.

LEMMA 1. Equation (2) has for fixed p a unique solution analytic in x and

in ζ on I x | £ 1 and \ζ\ £ 1, respectively, where x and ξ are complex vari-

ables. x

Proof. For fixed p , define

χ~ ξ)],

fj{x,op s

, ί,

Then

\ f ι ( * , O \ <M,

\fi(x,ξ)\ =

Hence, by induction,

- — £X Δ(x,*,p)fι(t9 ξ)dtόp s

\fj(x.t)\ <MN'~ι\x-ξ\J'1

consequently,

Λ (ί,

MN fx \dt\ = MN\x-ξ\.

/ - 2,3,4,...);

where Z<;(Λ;, ξ) is analytic in # and in ξ in | x \ £ 1 and | | £ 1, respectively.

By direct substitution into (2), we see that w{x9 ξ) is a solution.

To show uniqueness, consider

lrΓhe variables x and ξ will always be considered real, unless otherwise indicated,as here; in this case, as in subsequent cases, integration between complex limits, asin equation (2), may be taken along a straight line in the complex plane.

A THIRD ORDER IRREGULAR BOUNDARY VALUE PROBLEM 397

z(x, ξ) = ux(x, ξ) - u

2(x, ξ),

where uί(x, ξ) and u2(x, ξ) are solutions of (2). Clearly z(x, ξ) must satisfy

the equation

V > ζ / — ~~ "^^^^ / /\ ( ί c t r ) z [ t f i cLt *

3p *ίf

and for real Λ; and ^, z(x, f) is easily seen to satisfy the system2

L(z(x, ξ), λ) = 0, z{ξ, ξ) = z'{ξ, ξ) = z"{ξ, ξ) = 0.

Hence z(x, ξ) = 0 identically in x for any fixed ξ, for real x and £;this impliesz (χj ζ) - 0 identically for complex ac and ξ and completes the proof.

LEMMA 2. For reαί x and ξ, (2) is equivalent to the system

(2a) L(u(x,ξ),λ) = 0, u ( £ , £ ) = u ' ( £ , ξ) = 0, u " ( f , f ) = 3 p 2 .

Proof. Substitution in (2a) of u(x, ξ, p) as given by (2) shows that the

unique solution of (2) is a solution of (2a). However, for fixed ξ and p, (2a)

also has a unique solution. Clearly, these unique solutions must coincide, and

our proof is complete.

LEMMA 3. Let u(x, ξ, p) be a solution of (2). Then3

a) u(x, ξ, p) = eω3P E(x, ξ, p)

provided \ p\ is large enough p £ Sίf x _> ξ;

c

b) u{- ω2x, - cύ2ξ, p) = - ω 3 u(x, ξ, p ) ;

) " " ( 1 , 0 , p) = p 2 eω*PM(p),

where | A f ( p ) | >; m > 0, provided

p = - ^ - L - 77e1^ (0 < θ < τr/3),

2Unless otherwise indicated, the prime will always denote differentiation withrespect to the first indicated variable.

3Functions of p and other variables which are bounded for \ p\ sufficiently largewill be denoted by E( ).

398 GEORGE SEIFERT

for sufficiently large n.

Proof of a) . As in Lemma 2 of [3], p. 211, it follows that for p C Slt we

have

/ a \ ω3P(χ~ζ) r (ω2- ω3) p(χ-ξ) . t . Ί

u(x9 ξ, p) = e [ _ ω 3 _ ω 2 e 3 r +z(x,ξ,ρ)],

where | z (x, ξ, p) | < M for | p \ sufficiently large and x >_ ζ Hence

u{x,ξ,p)=eω3p{χ-t) E{x,ξ,p).

Proof of b ). Using (2), we have

u( - ω2x, - ω2ξ, p) = δ 3 [ - ω2p(x~ ^)]

1 /• ""^2 XJ e Δ{-ω2x, s, p) u(s,~ ω2ξ, p)ds

3p ~ω2s

= - ω 3 δ 3 [p(x-

But

J Δ v~ ω 2 *, - ω2t, p) u(-ω2t, - ω2ς, p)dt.3p s

ω 3

, p ) = δ 3 [p(Λ; - ί ) ] r(t)9

- ί ) ] ( - ω 2 p ( ί ) ) = ~ ω 3 Δ ( * , ί, p )

Hence

α ( - ω 2 ^ , - ω 2 ^ , p ) = - ω 3 δ 3 [ p ( « - ξ)]

fX Δ ( Λ , ί, p )

3p &

Multiplying this l a s t equation by — ω 2 , we have

A THIRD ORDER IRREGULAR BOUNDARY VALUE PROBLEM

where

z(χ, ξ» p) = - ω 2 w(- ω2x, - ω2ξ, p ) .

But by the uniqueness of the solutions of (2), we have

- ω2u(- ω2x, - ω2ξ, p) = " ( * , £ , p ) ;

upon multiplication by - ω 3 , this gives b).

Proof of c ). We have, from (2),

ω3 p

399

tt"(l,0,p) = p 2 [ S t ( p )

(ω2 ~ ωs)p 2

1 + e

for p £ S1# Let p = x + i'y, and define Φ (p) and rn by

E2(p)

Φ(p) = 1 + e andV3

respectively. With p = [3(r^ - x2)]ι/2, we have

I Φ ( p ) I > I 1 + e~P c o s ( V 3 " * ) | ,

provided p = rn e where 0 < θ < π/3, and will show that

— 1 Γne ~P cos (V 3 x) > for — < x < rn .

Since

cos (\/3 x) > 0 for rn ~ < Λ; £

it is clearly sufficient to show that

r- I Γne P cos ( γ 3 x) > for £ x < rn -

Accordingly, we note that for x in this interval, we have

400 GEORGE SEIFERT

e'P I c o s ( V 3 x ) \ < — N, provided N is sufficiently large. Taking N large enough we also

have

E2(p)< — for p = rne

iθ(0 < θ < τr/3).

Hence

E2(p)> | Φ ( P ) I -

E2(p)

2 T T

This completes the proof of the lemma.

By the formal series for f(x), we shall mean the series

Σ akUk(χ) where ak = £ f{x) vk(x)dx / f uk{x) vk{x) dx,

in which uk(x) and vk(x) are respectively the characteristic functions of the

system (1) and its adjoint corresponding to the characteristic value λ&.

LEMMA 4. The sum of the first n terms of the formal series for f(x) is given

by

ΔTTl ' n L

u(χ,0,p) f l

2πi X L " ' u " ( l , 0,p)

where σ(x) = J f(ζ) u(x, ξ, ρ)dξ, and γn is the arc of the p-plane given by

2n + 2 -ap = π eιθ, - τr/3 < θ < ιr/3,

V3"

A THIRD ORDER IRREGULAR BOUNDARY VALUE PROBLEM 4 0 1

the p integration proceeding in a counter-clockwise direction.

We omit the proof of this lemma, as its details almost duplicate the discus-

sion in [2], pp. 424-426- We point out, however, that Lemma 2 is required in

this proof.

LEMMA 5. The function σ{x) defined in the previous lemma satisfies the

equation

(3) σ(x) = fo

X f(ξ) S3[p(x-ξ)]dξ - - L jT* Δ(*. t, p) σ(t)df,

furthermore, σ(x) is its unique solution, is analytic on 0 £ x < 1, and can be

put into the form

σ(x) = u(x,09p) Ψχ(p) + Ψ a ( * , p),

where

3/(%) EΛX> p)Ψ 2 ( * , p ) = - + , E?(x,p) = p2 E2(x,p),

P p 2

provided f{x) = x2 φ{x3), where φ(z) is analytic on j z\ _< 1 .

Proof. Using (2) in the expression for σ{x), we obtain

1

= f* f(ξ) 83[p(x-ξ)Uξ- ^- f0

X ΔU,t,p)σ(ί)Λ

on changing the order of integration in the second integral. Uniqueness of the

solution σ{x) can be shown in the usual manner. (See tke proof of Lemma 1.)

We next substitute u(x, 0, p) Ψi(p) + Ψ2(#, p) into (3) for σ(x), and obtain

u{x,0,p)%(p) + Ψ 2 (* .P) = f *

- f* Δ(*, ί, p) w(ί, 0, p)c?ί / * Δ(Λ;, ί, p) Ψ2(ί, p)dt,3p ° 3p °

402 GEORGE SEIFERT

Using (2) with ^= 0, and subtracting the term u(x, 0, p) Ψi(p) from both sides,

we obtain

(4) %{χ,p)- f* f(ξ) δalpix-ξndξ-V^p) 83(px)

- — fo* Δ(Aί, t, p) %(

On integrating by parts twice, we obtain

Γ f(ξ) 83[p(x- ξ)]dξ = ¥ t l + p~2 f* f"(ξ) 82[p(x-ξ)]dξP 0

= - ^ + P"2 δ 3 (px) fy f"(ξ)eP£ dξ + £ 3 f"(ξ) ert dξ,P °

where y is a complex number to be determined later, and

fX F(t)dt -ω2eω2pX J"2* F(t)dt

pX Sω3% F{t)dt.

It is in this step that we use the form of f(x) as stated in the hypothesis of

this lemma; for the details, see [2], pp. 428-429.

We also have

fo

x A(χ,t,P) ψ 2 ( t , p ) Λ - i j£* ί , [ p U - 0 M 0 V2{t,p)dt

jf

δ 3<P*) *y t n

y r(t)ePι Ψ2(ί, p)dt + £ 3 r{t)ePι Ψ2(ί, p)dtP

δ3(px) fX

P(t)ePtΨ2(t,p)dt+


= 83(px) fj R{t)eP

tΨ

2(t,p)dt+ £

3/?(ί)e^ Ψ

2(t,p)ώ,

where R (t) = r( t )/p + p( ί ) , and where we have made use of the properties

of p(t) and r(t), and the fact that, from the form of Ψ 2 ( ί , p) in terms of u(x,

0, p) and Lemma 3, part b, we have

Ψ 2 ( - ω 2 ί , p) = - ω 3 Ψ 2 U, p ) .

Putting these results into equation (4), we obtain

δ 3

T 7

(px)' •

3 / (0

w-ij 're

1 ry

+ 3p" Jo

ePι dt - — £ 3 R (

)ePt dξ

{t)ePt Ψ2(ί, p)dt

t)ept Ψ2(ί, p)t/ί.

This equation will certainly be satisfied if

(5) Ψ 2 ( * , p ) - ^ ^ + — £3 i"U)ePtdt+ —P p 2 3p

and

-1- fj R(t)e^ Ψa(ί,

The proof of the existence of a unique solution ^2(χ9 p) °ί (5) will follow

along the lines of the corresponding proof in [2], provided we can show that

an expression of the form | £ 3 F(t)ePt dt\ is bounded for complex p and 0 _£

x £ 1 whenever | F (z ) \ is on | z \ £ 1 and we take y = - e~ι a r g p. For we have

^ x \F(t)\

F ( O | | e ^

404 GEORGE SEIFERT

f* \eP'\ \dt\

'ω2X \ePι\ \dt\ + \eω3pX\ f ' 0 * * ^ \ d t \ ] ,

f

+ \eω2pX

where | F ( z ) | < μ on \z\ £ 1; and since each integrand in this last ex-

pression assumes its maximum at its upper limit, we have

I £ 3 F(t) ePι dt\ < 6 μ .

We omit the rest of this existence proof. (See [2], pp.429-430.)

For the asymptotic form of Ψ2(#, p), we substitute

3/(*)Ψ2U, p) = + v(x, p)

P

into

(6)

For

m *

(5). We

fixed p

1

<

obtain

let m =

+ Ί 7

1

p2

-

max0<* 2μ2, where | £ 3 R(t)ept dt\ < μ2 Hence for such p we have

m < 2μχ/ I p I 2 , and it follows that v(#, p) = p*"2 Ex (x, p ) .

It remains to show that v"(x, p) = E2 (x, p). Differentiating (6), we have

(7) v'{x,p) = - i | £ 2 [ / ' ' ( 0 + R(t)f(t) + — Ex{t, p)]ePl dt\

E3(x, p)

where

Γ F(t)dt- ω3eω2pX

and we have used the fact that

\ E ί ( - ω 2 x > p ) \ p2 Ψ 2 ( ~ ω 2 * , p) -3/(-ω2%)

- P 2 ω 3 Ψ 2(*. p) - = \Ex{x,

< m2 .

We can also show, as before in the case of the £ 3 operator, that if | F (z) \

on | z | < 1, then | £ 2 F ( ί ) e ^ ί Λ

Differentiating (7), we obtain

—3p

where

Z.F(t)dt1

Γ F{t)dt+ eω2pX f~"2X F(t)dtJy Ίy

+ e " 3 ^ / " ^ F ( ί ) ώ ,

and we have used the fact that | E[ (- ω2x9 p) | = | £ί ( > p) | and that

E[{χ,p)\ = | p ^ ' U , p ) | < | p | M

for [p[ sufficiently large.

Hence v" (x, p) = E2 (x> p) s ince again | F ( z ) | £ μ for | z | £ 1 implies

I £j_ F(t)ept dt\ £ m l f and the proof of the lemma is complete,

3. Theorem. We proceed now to the proof of the following theorem.

THEOREM. // f(x) = x2φ(x3), where φ{z) is analytic on \z\ < 1, the

formal series for f(x) converges uniformly to f(x) on 0 < x < 1.

406 GEORGE SEIFERT

Proof. Since, for real x and ξ, u(x, ξ, p) is real for real p, by the principle

of reflection we have u{x9 ξ9 p*) = [u(x9 ξ9 p)]*. This implies that the inte-

grand in the expression for In(x) given in Lemma 4 takes on values for p on

)/ = γn n Sx which are the complex conjugates of those it takes on for p on

γ£ = γn n S2 It suffices, then, to consider only the p integration over )/. De-

noting the result by !£ (x), we have, by Lemmas 4 and 5,

liM - ΰi k u(x, 0, p), P)

u(x, 0, p)

u " ( l , 0, p)«"(1, 0, p)

and since, by Lemma 3, parts a) and c), we have

«U, 0, p) M

IPI

for p on y£ and n sufficiently large, we obtain

) E(x,P)

2

where

lim eή(x) = 0

uniformly in Λ This proves the theorem.

At the expense of brevity, this theorem clearly could be generalized to prob-

lems involving somewhat more complicated boundary conditions and somewhat

weaker analyticity conditions on f(x), p(x), and q(x); in connection with the

latter contention, see [2].

REFERENCES

1. A. H. Flax, Aeroelastic problems at supersonic speed, Proc. Second InternationalAeronautics Conference, International Aeronautical Society, New York, 1949, pp. 322-360.

2. L. E. Ward, A third order irregular boundary value problem and the associatedseries, Trans. Amer. Math. Soc. 34(1932), 417-434.

3. G. Seifert, A third order irregular boundary problem arising in aeroelastic wingtheory, Quart. Appl. Math. 9 (1951), 210- 218.

UNIVERSITY OF NEBRASKA

WELL-ORDERED SUBSETS AND

MAXIMAL MEMBERS OF ORDERED SETS

HERBERT E. VAUGHAN

1. Introduction. Kuratowski [2] and Mil gram [3] have given procedures for

avoiding the use of transfinite numbers in the proofs of many theorems. The

procedure of Kuratowski is based on results of Hessenberg [1] concerning the

existence of well-ordered subsets of ordered sets, and Milgram's is based on

a similar theorem [3, p. 23, Th. 1']. Although Milgram's theorem does not strictly

include Hessenberg's results, his procedure is more generally applicable than

is Kuratowski's. Milgram has given two proofs of his theorem, of which the

first (and simpler) utilizes transfinite numbers; the second, like Hessenberg's,

avoids their use by methods similar to those used by Zermelo in his second

proof of the well ordering theorem [6] Szele [4] has given a proof, patterned on

Zermelo's first proof of the well ordering theorem [5], that Zorn's lemma is a

consequence of the axiom of choice.

The purpose of this note is to show that Szele's proof can be modified in

such a way as to prove a common generalization (Theorem 1) of the theorems of

Hessenberg and Milgram. The proof is as simple as Milgram's first but, like his

second, makes no use of ordinal numbers. Furthermore, it can be arranged in

such a manner that Szele's theorem appears as an intermediate result almost as

simply as in [4].

2. A general result. We shall use the following notations. The relation <

orders M if it is transitive: a a < c; n is an upper (lower)

bound oi N \i p <_ n (n <_ p) for every p C N; n is a greatest ( least) mem-

ber of N if it is an upper (lower) bound of N and belongs to N; m is a maximal

member of M if m < n=ϊ>n < m for every n C M; a subset of M will be said to

be bounded if it has an upper bound; the symbol 'C* will denote proper in-

clusion, and Άf will denote the empty set.

The relation < well orders N if every nonempty subset of N has a unique

least member. If this is the case then < is transitive, antisymmetric: a a - by and connected: a Φ b = > a < b or b < a, on N If n C /V

Received November 5, 1951.

Pacific J. Math. 2 (1952), 407-412

407

408 HERBERT E. VAUGHAN

then Nn = {x \ x £ N and n < x \, and if n is not the greatest member of N then

n+ is the least member of \ x \ x £ N and x <_ ft }. The greatest member of Nn+ is

n.

THEOREM 1. Let M be ordered by <, a ζiU, and u a function which assigns

an upper bound to every bounded subset of M in such a way that: if N, P ζ^ M and

have a common greatest member then u(N) = u(P); while if Λ C N> P C, M, and

neither has a greatest member but both have the same nonempty set of upper

bounds, then u(N)=u(P); and, finally, u(Λ) = α. Let \i be the class of all

sets N C M each of which is well ordered by < and has the property that, for

every n £ N, we have n - u(Nn ). Let P be the class of all sets P C M each

of which has the property that, for every bounded set N C_P which contains no

upper bound of P, we have u(N) £ P, while if N C P and contains an upper

bound m of P, we have u{N) <^ m. Then the greatest member of ϊl and the least

member 0/P exist and are the same set, M(u). The least member of M(u) is a,

and if M(u) is bounded it has a greatest member m, and u(M (u)) <_m. A neces-

sary and sufficient condition that an m £ M(u) be the greatest member of M(u)

is that u({m\) < m.

REMARKS. If < is antisymmetric then r is the class of all sets P C_ M such

that, for each bounded set N C P, we have ι*(/V)£ P.Under the same hypothesis

we have: If M{u) is bounded, then u[M(u)] is its greatest member; and an

m £ M(u) is its greatest member if and only if u{\ m \) = m.

Milgram's theorem is a consequence of the theorem obtained from the above

by requiring that < be irreflexive and that u(N) = u(P) whenever N and P have

the same nonempty set of upper bounds, and by removing the requirement that

w(Λ) = α Theorem 1 then generalizes only the special case of Milgram's the-

orem which is obtained by adjoining 'u(A) = a* to the hypotheses of the latter.

However, this normalization places no restriction on the applicability of Theorem

1 and is introduced merely to simplify the statement and proof of this theorem.

Hessenberg's results follow from the special case of Theorem 1 which is ob-

tained by requiring that < be reflexive and antisymmetric, that if Λ C N C_ M

then sup W exists, and that there is a transformation f oί M in itself such that

for every m £ M we have m < f(m), and in terms of which u is defined as fol-

lows: For each N C M which has a greatest member n we have u(N) = f(n),

while if Λ C N C M and N has no greatest member, then u {N) = sup N.

3. Proof of Theorem 1. There exist nonempty sets which are members of

H, for example { a}, and a is the least member of every such set. If N9 P £ H

then one is a segment of the other. For if Nn = Pp is a common proper segment

of N and P then n = u(Nn) = u(Pp ) = p, and Nn u { n \ = Pp u { p } is a larger

WELL-ORDERED SUBSETS AND MAXIMAL MEMBERS OF ORDERED SETS 4 0 9

common segment of N and P. Hence the greatest common segment of N and P

(which exists, since it is merely the join of all common segments) is not a

proper segment of both of them and so must be either /V or P. It follows that the

join, M(u), of all members of ϊi is a member of H. If M(u) is bounded and

u [ M { u ) ] € . M ( u ) t h e n u [ M ( u ) ] i s t h e g r e a t e s t m e m b e r o f M. I f u[M ( u ) ] ψ

M(u) then M(u) u { u[M(u)] \ ft. ft. This can only be because, for some m G

M(u), we have u[M(u)] ^_m. Since < is transitive, m is the greatest member

of M.

If m G M(u) and is not the greatest member of M(u) then, since m is the

g r e a t e s t member o f M ( u ) m + , w e h a v e u({ m \) = u [ M { u ) m + ] - m + { m.

Suppose N C^M(u) and is bounded. Then N has an upper bound in M(u).

For if this were not the case then every upper bound of N would be an upper

bound of M(u) and the latter would have an upper bound, and consequently a

greatest member, contradicting the assumption that N has no upper bound in

M(u).

Now l e t n b e t h e l e a s t u p p e r b o u n d of N in M(u). If n G N t h e n e i t h e r n i s

t h e g r e a t e s t m e m b e r of M( u) a n d u(N) = u(M(u)) <_ n, or u(N) = u(M(u)Λ+) =

n+C M(u). If n ζj/ί N then either N = Λ and u(N) = a G M{u), or Λ C N and

neither N nor M(u)n has a greatest member but both have the same upper bounds

and u(N) = u[M(u)n] = n CM(u). Hence M(u) G P.

Finally, if P G P then M{u) C P. For suppose that m G M(u) such that

M(u)m C^P. Then M(u)m cannot contain an upper bound of P. For if n were

such an upper bound then, since P G P, it would follow that m = z*[M(zOm] < n,

contradicting the fact that, since n G M(u)m, we have m %_ n. Hence if

MU)m ^ ^ t h e n w β " [ Λ ί ( t t ) m ] CP. It follows by induction that M(u) C^P.

4. Corollaries. We note the following results.

COROLLARY I. Let u satisfy the additional restriction that any m G M

which is not maximal satisfies u ({ m \) s_ m. If M(u) is bounded, then u[M (u ) ]

is a maximal member of M.

Proof. If M(u) is bounded then, by the final result of first paragraph of the

preceding proof, it has a greatest member m, and u(\m\) _< m. The restr ict ion

on u then requires that m be maximal. But m <_ u[M(u)], so that u[M(u)] i s

a lso maximal.

COROLLARY 2. Let M have the property that if A C N C M, and N is

bounded, then N has a least upper bound; and let u satisfy the additional re-

strictions that if A C V C^M, and N is bounded but has no greatest member, then

u(N) is a least upper bound of N; and that if m, nζLM and m < n then u (\ m}) £

410 HERBERT E. VAUGHAN

u(\n\). If there exists a p C M such that a <^ p and u(\p\) £ p then M{u)

is bounded and u[M(u)] is a least such p (cf. [2, p . 83. Th. I I " ] ) .

Proof, Suppose a <_ p and u(\p\) <_ p . If m £1 M(u), not the greatest,

and m <_ p, then m+ - u({ m\) <_ u({ p}) <^p. If m £. M (u) and has no immedi-

ate predecessor in M(u), and p is an upper bound of M(u)m, then m = u(M(u)m)

£ p. Hence, by induction, p i s an upper bound of M(u). Hence M(u) has a

greatest member m9 with u(M(u)) <_ m <^p and u(\ u[M(u)]\) <^w({m}) £

m <_ u[M(u)] Consequently, u[M(u)] is a least such p.

5. On Zorn's lemma. We give now an alternative proof of Sze le ' s resul t :

THEOREM 2. The axiom of choice implies Zorn's lemma.

Proof. Let M be ordered by <, a £ M, and / a choice-function for the set

of all nonempty subsets of M. Define g as follows: g(M)-a; if N C M and

either has no least member or coincides with the set of its least members then

g(N) - f{N); if N C M and has a least member but does not coincide with the

set No of its least members then g(N) = f(N - iV0). Define u as follows:

u(N) = g(\x\x C M and (y) (y £ N = > y < *)}) . Then a ( Λ ) = α and if

yV, P C_ M and have the same nonempty set of upper bounds then u(N) - u(P).

If m is a nonmaximal member of M then the set of upper bounds of { m \ has a

least member m, but does not coincide with the set of its least members. Hence

u(\m\) is not a least upper bound of { m}, and u(\m\) jfc m. Since the hy-

potheses of Theorem 1 and Corollary 1 are satisfied, we see that if M(u) is

bounded then u[ώf ( u ) ] is a maximal member of M, and a _< u[M(u)].

From this, one obtains Zorn's lemma in the strong form (cf. [3, p 27, Th. 9]):

// M is ordered by <, and every subset of M which is well-ordered by < has an

upper bound, then for each a C M, there is a maximal member m of M such that

a £ m.

6. Remarks. Note that only the first paragraph of the proof of Theorem 1

is appealed to in the proof of Theorem 2, and that for this application the hy-

potheses on u can be simplified to the requirements that u(A) = a and that if

Λ/ and P have the same nonempty set of upper bounds then u(N) = u(P).

It may also be worthy of note that, in the first paragraph of the proof of

Theorem 1, the transitivity of < was used only in the last sentence. If < is

assumed to be antisymmetric (but not necessarily transitive) this sentence

might be replaced by: Since < is antisymmetric, we have m = u[M(u)~\ Up to

this point the only properties of u that have been used are that u(Λ) = a and

that u(N) is an upper bound of N. Appealing to the hypothesis that if N and P

WELL-ORDERED SUBSETS AND MAXIMAL MEMBERS OF ORDERED SETS 4 1 1

have a common greatest member then u(N) = u(P), we can conclude that

u({u[M(u)]\) = u(M{u)).

Now let an m G M be called 'weakly maximal' when, for every n C M, it

satisfies the following condition: if p _£ m = > p < n for every p ζl M, then

n < m. Clearly, a maximal member of Λί is weakly maximal, and if < is transitive

then the converse also holds. If < is antisymmetric then the phrase 'n < m'

can be replaced by 'n = mf . If, in Corollary 1, 'maximal' is replaced by

'weakly maximal' then the resulting statement is true not only if < is transi-

tive but also if it is assumed only to be antisymmetric.

Using the axiom of choice, one can define a function u which satisfies the

four conditions referred to in the two preceding paragraphs, as follows: u{A) — a;

if N has a greatest member m which is not weakly maximal then

u ( N ) = f ( \ x \x C M a n d x £ m a n d ( y ) ( ( y C M a n d y < m ) = > y < x ) \ ) ;

if N has a greatest member m which is weakly maximal then u (N) = m; if N is

bounded and nonempty and has no greatest member then

u(N) = f(\x\x C M a n d ( y ) (y € N = 4 > y < x)\).

Hence we have the following analogue of Zorn's lemma:

If < is an antisymmetric relation defined on M, and every subset of M which

is well-ordered by < has an upper bound, then, for each a £ M, there is a weakly

maximal member m of M such that a <_ m; that is, a £ m, and \{ n Φ m then

there is a p such that p _£ m and p f n.

7. Addendum. After this paper was submitted to the editors it was brought

to the author's attention that certain results similar to those stated in Theorem

1 had recently been published by Bourbaki [7]. He assumes that a set E is

given on which there is a transitive and antisymmetric relation £ and a trans-

formation / of E into itself such that: x £ f(x) for every x C E. In addition

a d E, His theorem asserts the existence of a set A which is the least of the

sets P C E such that a £ P; if x C P then f(x) C P; if Λ C N C P and N

has a least upper bound then this least upper bound belongs to E. The theorem

also asserts that _£ is connected on A. Bourbaki later proves that A is well

ordered by £ , and that A has a greatest member if and only if it has a least

upper bound.

It is interesting to note that Bourbaki's results follow from Theorem 1. For

the proof, let α^ be anything not in Ey and let M be E u {α^ }. Extend the re-

lation _£ and the function / to M by requiring that x < a^ for every x £1 £, and

that /(αfco ) = αoo. Define u as follows: u(A) = α; if Λ C N C_ Λί and is bounded,

4 1 2 HERBERT E. VAUGHAN

there are three cases: If N has a least upper bound m but no greatest member,

then u(N)=m; if N has a greatest member n then u(N) = f(n); if N has no

least upper bound then u(N) = α<χ,. With this definition it is easily shown that

M(u) - {cioo } is the desired set A.

REFERENCES

1. G. Hessenberg, Kettentheorie und Wohlordnung, J. Reine Angew. Math. 135(1909),81-133.

2. C. Kuratowski, One methode d'elimination des nombres transfinis des raison-nements mathe'matiques, Fund. Math. 3(1922), 76-108.

3. A. Milgram, Partially ordered sets, separating systems and inductiveness, Rep.Math. Colloquium, Second Series, Issue 1 (1939) 18-30.

4. T. Szele, On Zorn's lemma, Publ. Math. Debrecen 1(1950), 254-257.

5. E. Zermelo, B ewe is dass jede Menge wohlgeordnet werden kann, Math. Ann. 59

(1904), 514-516.

6. , Neuer Beweis fur die Wohlordnung, Math. Ann. 65(1908), 107-128.

7. N. Bourbaki, Sur le theoreme de Zorn, Arch. Math. 2(1949) 434-437.

UNIVERSITY OF ILLINOIS

AN OPTIMUM PROBLEM IN THE WEINSTEIN METHOD

FOR EIGENVALUES

H. F. WEINBERGER

1. Introduction. The method of Weinstein [l] gives upper bounds for the

eigenvalues λί >, λ'2 ^_ of the projection U into a space ® of a completely

continuous positive symmetric operator L in a Hubert space § with eigenvalues

^i 2l ^2 >.•••• These upper bounds are the eigenvalues λ^1' of the projection

of L into a space of finite index m,

(1) § Θ f Pi* " >Pm\f

where plf , pm are any vectors in the space

(2) φ = § 0 ® .

The chief part of the Weinstein method is the explicit determination of the

eigenvalues λ^m' in the space (1) in terms of the eigenvalues and eigenvectors

of L in ίρ . These satisfy

(3) λjf° > λ'n.

The values λ^™' will, of course, depend on the choice of the vectors (plf

• > Pm )• It is naturally desirable that the upper bound for a particular eigen-

value λ' should be as small as possible. This paper investigates how small it

can be made for given n and m by a proper choice of the constraint vectors

( P i » • ' " 9 Pm ) •

Because of the minimax principle, λ^m^ must satisfy

Our result is that the inequalities (3) and (4) are the only restrictions on the

smallness of λ . In other words, for given n and m, there exist vectors (p t ,

• * J Pm) such that the weaker of the inequalities (3) and (4) becomes an equality.

Received December 12, 1951. This work was sponsored by the Office of Naval Re-search.

Pacific J. Math. 2 (1952), 413-418413

4 1 4 H. F. WEINBERGER

2. The case of a single constraint. We first prove our result for the case of

the first intermediate problem, that is, for m = 1.

THEOREM 1. For any given n, there is a vector p in the space

(5) 5β - § 0 @

such that, if the projection of L into ίξ) Q { p \ has eigenvalues

> λ >

either

(6) λ»> = A;

or

according as λ^ O Γ λ n + 1 is larger.

Proof. If λ^ = λn, then (6) is satisfied for any p and there is nothing to

prove. Our theorem thus naturally splits into the two cases λ^ > λ^ L n^ι a n c ^

λ^ < λ^ + 1 , which we shall prove separately.

3. The case λn > λ'n >_ λ n + 1 Let the eigenvector of L' corresponding to

λ ' be α ' Its eigenvalue equation can be written in terms of the operator L a s

(8) Lu'n-X'nu'n=P,

where p is some vector in 5β. Let us assume for the moment that p is not a

null vector. Then (8) is an eigenvalue equation for the projection of L into ® but

not for L. Any eigenvector of L corresponding to the eigenvalue λ^ must, be-

cause of (8), be orthogonal to p and hence must belong to § Q \ p } Thus, the

multiplicity of λ^ as an eigenvalue of the projection of L into § Q { p } is one

greater than its multiplicity as an eigenvalue of L. Let the latter be r > 0. If

r = 0, then λ^ > ^n+l9 and λ^ must be λ ^ by the minimax principle. If r _> 1,

then λn > λΛ + 1 = ••• = λ n + Γ > λ π + r + 1 , and the minimax principle gives

(9) λ£.\ >λn>x'n> λ π + 1 > λ π + r + 1 >

Thus, since the multiplicity of λ^ in § Q { p } is r + 1, we must have

AN OPTIMUM PROBLEM IN THE WEINSTEIN METHOD FOR EIGENVALUES 4 1 5

do) ^ ;

so that the vector p in (8) has the property stated in our theorem.

If λ^ = λ Λ + 1 , it is possible that the vector p in (8) is a null vector. This

means that the eigenvector of L' corresponding to λ^ is a lso an eigenvector of

L. Suppose that the same is also true of the eigenvalues λ^ + 1 , ••• , λ^ + s but

not of λ ' + We then consider the projections L and L' into

{ } and @ e

respectively, and call their eigenvalues λ^ and λ . Then L' has the same eigen-

values as Z/, except that the eigenvalues λ^, ••• , λ^ + s are removed. The

same is true of L and L. Then

If there is a vector p in 5β so that the π-th eigenvalue of L in

is at most λ ' , then, because of (11), the rc-th eigenvalue of L in § Q \ p

λ^ and equation (6) in our theorem will be proved. Now if

then, since by definition of s the eigenvector of 1/ corresponding to λ^ + s i s

not an eigenvector of L, the existence of such a vector p follows from the first

part of this paragraph. If, on the other hand, we have

the existence of this vector p will be assured by the results of the next para-

graph.

A final possibility* is that there is no integer s such that the eigenvector

wΛ + s of U is not also an eigenvector of L. In other words, all but the first

n - 1 eigenvectors of L' are also eigenvectors of L. Then, since λ^ = λ/ι + 1> the

vectors u'9 ••• , M^ are the only eigenvectors of Z/ which are not orthogonal

to ul9 U2J , Un Therefore there is one linear combination p of u2, ••• , un

which is orthogonal to all eigenvectors of L' and hence belongs to Sβ. There can

*This possibility was pointed out by C. Arf in the course of an alternative proof ofthe results here presented.

be at most n ~ 1 eigenvectors of the projection of L into § © ί p 5 which are

not orthogonal to ul9 -•• , un. Therefore, the rc-th eigenvalue of this projection

is λ^+1 = λ^, and both equalities (6) and (7) hold.

4. The case λ^ < λ n + 1 We now show that if λ^ < λ n + 1 then the equation

(7) can be made to hold. This will be done by induction. We first replace the

space 5β by a finite space. Since L is completely continuous it follows that

λ m —>0 as m—»oo; therefore there is an integer m such that

It has been shown by the author [1,2] that if we let pi be the projection in

ty of Uj, then the eigenvalues λ^1' of L in ίρ © { pl9 , pm \ satisfy

( i s ) λ^m) < Λ; +

Combining this with (14), we obtain

Thus, it will suffice to show that if the inequality (16) holds where λ^m' is

the 7i-th eigenvalue of L in a space ξ> © { pl9 , pm }, then there is a linear

combination p of the vectors pl9 ••• , pm such that the τι-th eigenvalue of L in

§ © { P 1 is λ Λ + 1 Our induction proof consists of showing that if (16) holds for

m > 1 then there is a linear combination p ' of pm-χ and pm such that the n-th

eigenvalue of L in ίρ © { pϊ9 , P m _ 2 , p ' ] is at most λ Λ + If λ^m~1^ £ λ Λ + 1 ,

this is obviously true, for we must only take p ' = P m - 1 Thus, we need to ex-

amine only the case

(17) λ (

n

m ~ 2 ) > λim~ι) > λM + > λ < m ) .n — n TIT i — n

Since, by the minimax theorem,

(18) λ^+1 > λ ( ^ 2 ) ,

our induction step will be proved if we find p so that the rc-th eigenvalue of L

in § © {pi, ••• , Pm-2> P'J i s e c l u al to either λ^m' or λ j^~ 2 ' . In other words,

the induction step is just Theorem 1 in the special case in which 5β is a 2-

space.

Thus if λ^m' > ^nΊi t n e induction is proved by the results of §3. Note

that in the case of a common eigenvector where one had to reduce the proof

AN OPTIMUM PROBLEM IN THE WEINSTEIN METHOD FOR EIGENVALUES 4 1 7

in §3 to the proof of this section, the reduction is to the case λ^m' < λj^~ ,

which will now be treated.

If λ^m' < λ ^ ~ 2 , we must construct a linear combination p ' of p m - ι and pm

so that ^l\ is t n e rc-th eigenvalue of L in § 0 \pl9 ••• , Pm-2> P'^ ^° °

this, we take for p ' the linear combination of P m . ι and pm which is orthogonal

to the eigenvector corresponding to λj™~2'. Then λ ^ ~ 2 ' is an eigenvalue of

L in ξ) 0 { p l f , Pm_2> p ' }• By the minimax principle, the (ra + l)-st eigen-

value in this space is at most λ ^ < λ ^ ~ 2 \ Therefore λ ^ 2 ^ must be the rc-th

eigenvalue in this space, and p ' has the desired property.

Thus, our induction step is proved and Theorem 1 has been shown to hold

in all possible cases.

5. The general intermediate problem. We are now in a position to prove the

more general result announced in the introduction.

THEOREM 2. For any fixed integers m and n, there are vectors pl9 •• , pm

in P which, if used as constraints in the n-th intermediate problem, yield either

(19) λ l m ) = λ'

or

(20) λ? 0 -*„+„•

Proof. We first prove the possibility of the equality (20) when

According to Theorem 1 with n + m — 1 substituted for rc, there is a vector

p t such that

(22) λ ^ = λn+m.

We then apply Theorem 1 to the projection of L into § 0 ί Pi ! to assert

the existence of a vector p2 such that

(23)

This process is repeated until the equality (20) is obtained. Inequality (21)

assures us that the equality (7) of Theorem 1 will always be attainable.

If λ^+m <_ λ^, then there is an integer I < m such that

(24) K+l<K< W i

We shall show that there are I vectors p l 5 , p, for which

(25) λ<f> = K

The equality (19) will then hold for any m - 1 vectors Pj + 1> , Pm appended to

the first I.

Since λ^ >. >. λ^+ j_j, we can proceed as in the proof of (20) to show

that there are I - 1 vectors p t , , p^_ χ for which

(26)

We now apply Theorem 1 to L in ξ> © { p x , , P^χ ί According to (24) and

(26) it is the equality (6) which can be made to hold by a constraint p^. We thus

obtain (25), and Theorem 2 is proved.

R E F E R E N C E S

1. H. F. Weinberger, An error estimate for the method of Weinstein, Abstract of apaper presented at the Minneapolis meeting of the American Mathematical Society, 1951.

2. , An error estimate for the Weinstein method for eigenvalues. Submittedto Proc. Amer. Math. Soc.

3. A. Weinstein, Etude des spectres des equations aux derivees partielles de la

theorie des plaques elastiques, Memorial des Sciences Mathematiques, 88, Gauthier-Villars, Paris, 1937.

4. N. Aronszajn, Rayleigh-Ritz and A. Weinstein methods for approximation of eigen-values. Proc. Nat. Acad. Sci. 31 (1948) 474-480 and 594-601.

5. N. Aronszajn and A. Weinstein, On the unified theory of eigenvalues of plates andmembranes. Amer. J. Math. 64(1942) 623-645.

INSTITUTE FOR FLUID DYNAMICS AND A P P L I E D MATHEMATICS

UNIVERSITY OF MARYLAND

NOTE ON FOURIER ANALYSIS XXXI:CESARO SUMMABILITY OF FOURIER SERIES

S H I G E K I YANO

1. Introduction. M. Jacob [1] proved that if the series

Σ \ a \ o \ n \ U \ OC x . )

converges then the trigonometrical series ^ ^ = 1 (an c o s n x + bn sin nx) is

summable (C, — Ot) almost everywhere. This result is equivalent to the pro-

position that if the series Σ ° ° = 1 (an cos nx + bn sin nx) is the Fourier series

of a square-in tegrable function, then the series ^°° = 1 (an c o s n + ^n s^n nx)n a

is summable (C, — a ) almost everywhere.

Considering the analogue of this theorem for integrable functions, we shall

prove here the following:

T H E O R E M I . 1 / /

aQ ™ A0(x) °°(1) + y, (an cos nx + bn sin nx) = + V An{x)

2 ^ 2 ^

ί's ίAe Fourier series of an integrable function f(x), then the series

oo

(2) «o + Σ (an c o s Λ Λ ; + ^n s i n nx)τΓa (0 < α < l )

αnc? iίs conjugate series

( 3 ) ^ ( α n sin TIΛ; — 6Π cos nx)n a

f. A. Zygmund has pointed out to the author that Theorem 1 can be obtaineddirectly from known results.

Received June 22, 1950, and, in revised form, October 20, 1951.

Pacific J. Math. 2 (1952), 419-429419

4 2 0 SHIGEKI YANO

are summable ( C, — Cί) almost everywhere,

2. Lemmas. For the proof we need the following lemmas.

LEMMA l . //

(4) H(

n

a> (x) = 1 + Σ k'a cos fee, /£•> (x) = 2 k'a s i n kx

7 1 = 1 7 1 = 1

( 0 < α < 1 ) ,

then

( 5 ) |ff<β> ( * ) | < / ί β * β ~ ι , | / ^ α ) (x)\ < Baxa~ι {0<x<π),

where Aa, β α > . . . are constants which depend only on α and which may be

different in different instances.

The proof of this lemma concerning the cosine series is given by Salem and

Zygmund [2], and the part concerning the sine series can be proved similarly.

LEMMA 2. // 0 < α < 1 and

then

(6)e

ikx

A;=o

α - i( u = 0 , 1 , 2 , •••

0 < | * | < 77).

Lemma 2 is a well-known result of M. Riesz. (Indeed, the constant on the

right side of (6) can be replaced by 2; however, the inequality in the above form

can easily be derived from Lemma 1.)

Let us denote the (C,-Oί)-means of the series (2) and (3) by Λ^α) (f x)

and N^a) (f x), respectively. Then, setting AΓα = 1 for k = 0, we have

Akί*)

FOURIER ANALYSIS XXXI: 421

= 1 f_π f{x + t) —! cos kt at

2

π ~π

dt>

where

(7)

Similarly

(8)

where

(9)

NW (f;χ) = - / _ w fix + t) N<*> (t) dt,

( ί )

LEMMA 3. For 0 < α < 1, we have

(10)0 < ί < 77)

Proof. From ( 7 ) , we have

i in/ 2]

(0 = c o s C 0 S

k=o

(11)= Pn + Qn,

say. Using Abel's transformation, we get

(12) Pn =1

A;=0

422 SHIGEKI YANO

By Lemma 1 we obtain

11 1 1

(13)

< Knata'

ίn/2]-ί

\An-k I +A°-1 Λn-lnn-tn/2]

< A t— a

Now, using Abel's transformation again, we have

(14) Qn =A<n-«>

where

Thus

n-l

j * c o s

C O S

ΔAΓα= A " α - (jfc + l Γ

m< ncos

k=[n/2]

(15)

[n/2] + l < m< πΣ *«

+ n

Since

Σ 4-J? c o s kt

Σ Λn-h-rikt

e"int Σ Ak~a) e ί k t

k=n-m

we have, by Lemma 2,

(16) Σ <Ώ> cos kt/c=o

Aa\l-

Therefore, from (15) and (16),

(17) \Qn\ £ Aata~l.

Combining (11), (13), and (17), we have the first inequality of (10). The

second inequality can be proved similarly.

3. Proof of Theorem 1. We proceed now to the proof of Theorem 1. From

(7), we have

dt

By virtue of Lemma 3, we get

|/V<α> (f;x)\ < Aaf_l\f(x + O | | ί | α " 1 dt,

whence

J Γ s u p | / V < α > (f;x)\ dx < A a J Γ dx f _ π \f(χ + t)\ I t l * " 1 dtn π

(18) \ f ( x + t ) \ d t

Similarly we have

(19) sup

\f(x)\ dx

dx < Ba £ \{{x)\ dx.

From these maximal inequalities we can easily deduce the conclusions of

Theorem 1. For example, we can proceed as follows. Let e > 0 be arbitrary, and

let f(x) - fι (x) + f2 (x)i where f\{x) is a trigonometrical polynomial and

J | / 2 ( Λ ; ) | dx < e/2Aa, Aa being the constant which appears in the right~"TT

424 SHIGEKI YANO

side of (18). By / (x) we denote the function defined by ] ζ ^ = 0 An{x)n α (this

series converges almost everywhere); then

(20) /*(*) = /*(*) + /•*(*),

where f*(x) and /2*(#) are determined by /\ (x) and f2 (x), respectively, in the

same way as / (x) is determined by f{x)

From (18) we have

(21) J_l sup |/V<«) if2ix)\ dx < A a J_l \fa(x)\ dx < e / 2 ,

and a fortiori,

(22) J_l \f*(x)\ dx< e/2.

From

(23) /V<α> (/;*) - /*(*) = N^ (fιiX) - f*(χ) + Nf> (f3ix) - f*(x)

we get

(24) l i m s u p 1/V<α> ( / , * ) - f * ( x ) \ < s u p | N^ {fa;x)\ + \f*(x)\,n—* oo n

whence, by (21) and (22),

(25) jΓlimβup |^ α > (f x) - f*(x)\ dx < e/2 + e/2 = e.n—*oo

Therefore it follows that /V<α) (/,x) —-> f*(x) almost everywhere; this implies

the validity of the first part of Theorem 1. The proof for conjugate series is

analogous,

4. On multiple Fourier series. Theorem 1 can be extended to multiple Fourier

series. For simplicity we shall state the result for the case of double Fourier

series.

Let f(x,y) be an integrable function periodic with period 2 77 in each vari-

able, and let its Fourier series be

(26) f(χ,y) ~ Σ AmΛ*>y),mth=o

FOURIER ANALYSIS XXXI: 4 2 5

where

A00(x,γ) = — α 0 0 , AOn(x,γ) = — (aQn cos nγ + bOn sin nγ),T* Zt

1Amo\χ>y> = " " (αmo c o s m * + /no s*n m ^)>

(27)i m n ( x , y ) = amn cos mΛ; cos 717 + bmn cos m% sin nγ

and

(28) αmΓι = J l / /(Λ:, 7) cos mΛ; cos τιy ώcίy (m, /ι = 0,1, 2, ) ,

and similarly for bmn , c m π and c?mn .

We shall say that the double series ^ ^ w = Q Amn is summable (C,(X,β)

if the ( C, α , /S )- means

- m n

L* Z* m-/ π-/c //c7=0 £=o

of ΣAn

Undei

THEOREM 2. If f(x,γ) is integrable, then the series

lmn converge.

Under these definitions we have:

(29) 2 Amn(x,y)m~an~β (0 < α < 1, 0 < jS < 1)

is summable (C, — (X, — β) almost everγwhere. A similar result holds for its

conjugate series.

Proof. The (C, - α, - β )-means ^ ^ β ) (f;x,γ) of the series (29) can be

written as follows:

(30) ^ β ) (f;χ,y) = — J Γ /_" / ( * + u, y + t;- -77 v ~ 7 7

7Γ2

4 2 6 SHIGEKI YANO

where Λ^α* (t) is the same as in (7) Following the line of proof of Theorem 1,

we obtain the result.

5. On the capacity of sets. Generalizing a result of A. Beurling, Salem and

Zygmund [2] proved the following theorem:

// the series

oo

Σ K2 + K) »α ( 0 < α < 1)n=\

converges, then the trigonometrical series ^ n ^ i (α Λ cos nx + bn sin nx) is

convergent except possibly on a set of (1 - &)— capacity zero.

We shall here prove the L t — analogue of this theorem.

THEOREM 3 Under the same assumption as in Theorem 1, the set E of the

points where the series (2) is not summable (C, — OC) is of (1 — CC) — capacity

zero. A similar result hold for the conjugate series (3).

REMARKS, ( a ) Theorem 1 is, of course, implied by Theorem 2. (b) For

the notion of capacity and other definitions, the reader is referred to the paper

[2] of Salem and Zygmund.

To prove Theorem 2 we need the following lemmas.

LEMMA 4. / /

= y n c o s nχy J v ^ ) = X n sin nx ( 0 < OC < 1 ) ,

then

f](a-) (^) ^ Λ;α~x / ( α ) (Λ;) ^ xa~ι (x > + 0 ) .

This is known (see, for example, [3; p 116]).

LEMMA 5. // a set E is of positive α -capacity (0 < OC < 1), then there

exists a positive distribution μ concentrated on E such that if the Fourier-

Stielties series of μ(x) is denoted by

1dμ(x) ~ / + V ( CCn cos nx + βn sin nx),

2π J~t

then the series

^ (QLn QOS nx + βn sin nx) n~ι*a

n-i

and its conjugate series

oo

]Γ ( an sin rc% - βn cos rcΛ;)rc~ι+α

are the Fourier series of bounded functions.

Lemma 5 is due to Salem and Zygmund [2j.

To prove Theorem 3, let us assume that the set E is of positive (1 — OC)-

capacity. Then by Lemma 5 we can find a positive distribution μ concentrated

on E such that if

1dμ(x) ~ + Σ (®>n c o s n x + βn s ι n nx) >

2 τ τ » = i

then the series Σ°°= ( &n c o s nx + βn s ι n nx)n a is the Fourier series of a

bounded function.

Since it is easy to verify by Lemmas 1 and 4 that

(31) \N^ ( 0 | £ Λ ta~ι < Λα//(α> ( 0 + Ba,

we h a v e , by (17),

U Z J | ;v n 1 / , Λ ) | < — I | / ( ί ) | /v; ; (Λ; - 0 | at7T

Integrating both sides of (22), we have

4 2 8 SHIGEKI YANO

y sup \N^ (f-,x)\ dμ(x)n

(33) < Λaf2π d μ { x ) fπ | / ( ί ) | / / ( α ) (x - t ) dt + B a fπ \f(t

< ^α / π | / ( 0 | Λ Γ27T ff(α) (^ - 0 dμ{x) + Ba fπ | / ( ί ) | dt.

By a well-known theorem, the Fourier series of

π °

is

1 + 2_, ( ^"Π cos nί + βγi sin n£) rι ,

which is the Fourier series of a bounded function. Iίence for almost all t we have

I / o

2 7 7 / / W (x - t) dμ(x)\ < A/(μ),

M{μ) being a constant depending only on the distribution μ. Therefore, from

(33), we have

fo s u p | Λ ' f > ( / ; * ) | d μ ( x ) < A J l ( μ ) /_" | / ( 0 | dt + B a f^ \ f ( t ) \ d t

(34)

< Ca(μ) J^ \((x)\ dx,

where C α (μ) is a constant depending only on CC and on the distribution μ.

Using this maximal inequality (34) and following the proof of Theorem 1,

we s e e 2 that

2We must first prove that the set of points where the series ^ ^ = 0 An(x) n α does

not converge is of μ-measure zero. This can be done by the maximal inequality

f 2 7 T s u p I 1 A k ( x ) k a \ d μ ( x ) < A { μ ) Γ \ f { x ) \ d x ,J0 n jt=o π

which is a simple consequence of (34), by the same argument as in [3? p 254] .

NOTE ON FOURIER ANALYSIS XXXI: 4 2 9

/o

27T lira sup |/V<α> (f,x) - f*(x)\ dμ{x) = 0,n —*oo

which implies that E is of μ-measure zero, contrary to the hypothesis that μ is

concentrated on E9 ( jL dμ(x) — 1).

For the conjugate series (3) an analogous argument can be applied, and

Theorem 3 is proved completely.

REFERENCES

1. M. Jacob, Uber die Verallgemeinerung einiger Theoreme von Hardy in der Theorieder Fourierschen Reihen, Proq. London Math. Soc. 26(1927), 470-492.

2. R. Salem, and A. Zygmund, Capacity of sets and Fourier series, Trans. Amer.Math. Soc. 59(1946), 23-41.

3. A. Zγgmunό, Theory of trigonometrical series, Warsaw, 1935.

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Pacific Journal of MathematicsVol. 2, No. 3 March, 1952

Lars V. Ahlfors, Remarks on the Neumann-Poincaré integral equation . . . . . 271Leonard P. Burton, Oscillation theorems for the solutions of linear,

nonhomogeneous, second-order differential systems . . . . . . . . . . . . . . . . . 281Paul Civin, Multiplicative closure and the Walsh functions . . . . . . . . . . . . . . . . 291James Michael Gardner Fell and Alfred Tarski, On algebras whose factor

algebras are Boolean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Paul Joseph Kelly and Lowell J. Paige, Symmetric perpendicularity in

Hilbert geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319G. Kurepa, On a characteristic property of finite sets . . . . . . . . . . . . . . . . . . . . . 323Joseph Lehner, A diophantine property of the Fuchsian groups . . . . . . . . . . . . 327Donald Alan Norton, Groups of orthogonal row-latin squares . . . . . . . . . . . . . 335R. S. Phillips, On the generation of semigroups of linear operators . . . . . . . . 343G. Piranian, Uniformly accessible Jordan curves through large sets of

relative harmonic measure zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371C. T. Rajagopal, Note on some Tauberian theorems of O. SzÃ¡sz . . . . . . . . . . . 377Halsey Lawrence Royden, Jr., A modification of the Neumann-Poincaré

method for multiply connected regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385George H. Seifert, A third order irregular boundary value problem and the

associated series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395Herbert E. Vaughan, Well-ordered subsets and maximal members of ordered

sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Hans F. Weinberger, An optimum problem in the Weinstein method for

eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413Shigeki Yano, Note on Fourier analysis. XXXI. Cesàro summability of

Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

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