MSP · 1BDJGJD +PVSOBM PG.BUIFNBUJDT INTHISISSUE— Tom M. (Mike) Apostol, On the Lerch zeta...

PacificJournal ofMathematics

IN THIS ISSUE—

Tom M. (Mike) Apostol, On the Lerch zeta function . . . . . . . . . . . . . . . 161Ross A. Beaumont and Herbert S. Zuckerman, A characterization of

the subgroups of the additive rationals . . . . . . . . . . . . . . . . . . . . . . . 169Richard Bellman and Theodore Edward Harris, Recurrence times for

the Ehrenfest model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Stephen P.L. Diliberto and Ernst Gabor Straus, On the

approximation of a function of several variables by the sum offunctions of fewer variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Isidore Isaac Hirschman, Jr. and D. V. Widder, Convolutiontransforms with complex kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Irving Kaplansky, A theorem on rings of operators . . . . . . . . . . . . . . . . 227W. Karush, An iterative method for finding characteristic vectors of

a symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Henry B. Mann, On the number of integers in the sum of two sets of

positive integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249William H. Mills, A theorem on the representation theory of Jordan

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Tibor Radó, An approach to singular homology theory . . . . . . . . . . . . . 265Otto Szász, On some trigonometric transforms . . . . . . . . . . . . . . . . . . . . 291James G. Wendel, On isometric isomorphism of group algebras . . . . 305George Milton Wing, On the L p theory of Hankel transforms . . . . . . . 313

Vol. 1, No. 2 December, 1951

PACIFIC JOURNAL OF MATHEMATICS

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COPYRIGHT 1951 BY PACIFIC JOURNAL OF MATHEMATICS

ON THE LERCH ZETA FUNCTION

T. M. APOSTOL

l Introduction. The function φ(x9a9 s), defined for Hs > 1, x real, a ψ nega-

tive integer or zero, by the series

o° 2nπix

(1.1) φ (x,ass) = Σ 1 7 y '

was investigated by Lipschitz [4; 5], and Lerch [3]. By use of the classic

method of Riemann, φ{x, α, s) can be extended to the whole s-plane by means of

the contour integral

1 Λ zs"1eaz

α.2) «•".•)/where the path C is a loop which begins at —-00 , encircles the origin once in the

positive direction, and returns to — 00 . Since I(x9a, s) is an entire function of s,

and we have

d 3 ) φ(x,a,s)=Γ{l-s)l(x,a,s),

this equation provides the analytic continuation of φ. For integer values of x,

φ(x,a,s) is a meromorphic function (the Hurwitz zeta function) with only a simple

pole at s — 1. For nonintegral x it becomes an entire function of s. For 0 < x < 1,

0 < a < 1, we have the functional equation

(1-4) φ{x, a,l-s)

first given by Lerch, whose proof follows the lines of the first Riemann proof of

the functional equation for ζ(s) and uses Cauchy's theorem in connection with the

contour integral (1.2).

Received March 4, 1951.

Pacific J. Math. 1 (1951), 161-167.

161

162 T. M. APOSTOL

In the present paper, §2 contains a proof of (1.4) based on the transformation

theory of theta-functions. This proof is of particular interest because the usual

approach (Riemann's second method) does not lead to the functional equation

(1.4) as might be expected but to a different functional relationship (equation (2.4)

below). Further properties of φ(x9α,s), having no analogue in the case of ζ(s),

are needed to carry this method throu'gh to obtain (1.4).

In §3 we evaluate the function φ(xf α, s) for negative integer values of s .

These results are expressible in closed form by means of a sequence of functions

βn{a9e2πix) which are polynomials in a and rational functions in e2Ίίιx. These

functions are closely related to Bernoulli polynomials; their basic properties also

are developed here.

2. Functional Equation for φ(x9 a, s ) . The theta-f unction

00

^3 (y lτ) = Σ exp (πin2r + 2iny)n=-oo

has the transformation formula [6,p.475]

If we let

θ(x,a, z) = exp(—ττa2z)&3(πx+πiaz\iz) = Σ exp(2nπix — πz(a + n)2) ,n=-oo

then we have the functional equation

(2.1) θ(a,~x,l/z)= [exp(2ττiax)]zi/2θ(χ,a,z).

The key to Riemann's second method is the formal identity

(2.2) π-°/*Γ(s/2) Σ «n/n's/2 = Γ ^ " 1 Σ α nexp(-τ,z/ n)dz.n = l

Taking first an = exp[(2mU ~l)x)] , fn = U - 1 + a)2 in (2.2), and then

an = exp(— 2τrinx), fn — (n — a)2 , we obtain

ON THE LERCH ZETA FUNCTION 163

(2.3) 7 τ - s / 2 Γ ( s / 2 ) {φ{x,a,s) + exp(-2πίx) φ(-x,l - a,s)}

In the second integral in (2.3) we apply (2.1) and replace z by 1/z. Denoting the

expression in braces by K(x9α9s), replacing 5 by 1 — s, re by ~ o , α by x, using

θ{ — α,x9 z) — θ(αf ~xf z), and the relation

( ^ p ) 2(2τrΓs

c o s (τrs/2)Γ(s) ,

we are led to

(2.4) Λ(x,α,l - s) = 2(2τ7)"5 cos (ττs/2)Γ(s) exp(~2πίαx) A ( - α , x,s) .

Thus Riemann's method gives us a functional equation for Λ instead of (1.4). At

this point we introduce the differential-difference equations satisfied by φ ,

namely:

α,s) , ,= —sφ\x,α,s + 1;

σα

and

(2.6) h 2τfiaφ\xiays) — 2τri φ{x, α, s — l) .

'σx

The first of these follows at once from (1.1). To obtain (2.6) we first write

exp[2πi(n + a)x~\φ(x,α,s) = exp( — 2πiαx)

(n + α)s

before differentiating with respect to x. The equations hold for all s by analytic

continuation.

The proof of (1.4) as a consequence of (2.4) now proceeds as follows. We differ-

entiate both sides of (2.4) with respect to the variable α, using (2.5) on the left

and (2.6) on the right, and replace s by s + 1 in the resulting equation. This

164 T. M. APOSTOL

leads to the relation

Φ{x, α, 1 —s) —exp(— 2ττix) φ(— x,\ —α, 1 —s)

= 2i(2π)"s sin (πs/2)Γ(s)

X [exp(~2τriαΛ:)φ(— α, %, S ) — exp(~ 2πia(l— x))φ(a, 1— x,s)] .

Adding this equation to (2.4) gives the desired relation (1.4).

This method has already been used by N. J. Fine [ l ] to derive the functional

equation of the Hurwitz zeta function. Fine's proof uses (2.5) with x = 0. In our

proof of (1.4) it is essential that x φ 0 since we have occasion to interchange the

variables x and α, and φ{x9a,s) is not regular for a — 0 hence Fine's proof is

not a special case of ours. Furthermore, putting x = 0 in (1.4) does not yield the

Hurwitz functional equation, although this can be obtained from (1.4) as shown

elsewhere by the author.

3. Evaluation of φ{x, a, — n). If x is an integer, then φ{x, a9 s) reduces to the

Hurwitz zeta function ζ(s,a) whose properties are well known [6,pp.265-279] .

For nonintegral x the analytic character of φ is quite different from that of ζ ( s , α ) ,

and in what follows we assume that x is not an integer.

The relation (2.6) can be used to compute recursively the values of φ(x, α, s)

for s = —1, —2, — 3, . As a starting point we compute the value at s = 0 by

substituting in (1.2). The value of the integral reduces to the residue of the

integrand at z = 0 and gives us

Φ(x,a,0) = -. - r = (i/2) cot πx + 1/2 .1 "" exp(27T ix )

Using (2.6) we obtain

0 ( * , α , - l ) = (α/2)(i cot 77* + l) - (l/4) csc2πx ,

φ(x,a,-2) = (α2/2)(i cot πx + 1/4) - {a/2) csc2rrx - (i/4) cot πx csc2πx .

If we put 5 = — 71 in (1.2) and use Cauchy's residue theorem we obtain, for

n > 0, the relation

φ\x9a,—n) =


where βn{a, α) is defined by the generating function

(3.1) , _ί^_. j«£«i . . .α e ' - l „.„ »!

When OC = 1, βn(af(χ) is the Bernoulli polynomial Bn(a) For our purposes we

assume Ot ψ 1, and in the remainder of this section we give the main properties of

the functions βn(a,(X).

Writing βn(a) instead of βn(0, α) we obtain from (3.1):

(3.2) βn(a, α) = £ lϊ) βkfa) an~k (n > 0) ,

from which we see that the functions βn(a9 OC) are polynomials in the variable α.

The defining equation (3.1) also leads to the difference equation

(3-3) aβn(a + l,α) -βn(a,a) =nan~1 (n > l) .

Taking α = Owe obtain, for n — 1, the relation

(3.4) 0LβΛl,0i) = 1 +j81(α)

while for ^ > 2 we have

(3.5) α/3 B ( i ,α)=y8 B (α) .

Putting α — 1 in (3.2) now allows us to compute the functions βn(CL) recursively

by means of

(3.6) /3n(l,α) = Σ (n\βk{o>)fe=o ' '

and (3.4), (3.5). From (3.1) we obtain /30(α) = 0; the next few functions are found

to be:

3α(α

u - 1 (cc- l ) 2 (&.-1)

_ , , 4α(ct2 + 4α + l) 5α(α3 + llα2 + llα + l)β4(a) = — , /35(α ) = —

( α - 1 ) 4 (α — l ) 5

166 T. M. APQSTOL

6α(α4 + 26α3 + 66α2 + 26α

(α-l)6

The general formula is

n-1(3.7)

where &jj are Stirling numbers of the second kind defined by

J 0 f e

i I

with

0 n = ( Δ ^ Λ ) X = O , Δ > 0 π = 0 i f j > n , Δ ° 0 ° = 1 ,

in the usual notation of finite differences. (A short table of Stirling numbers is

given in [2].)

To prove (3.7) we put

QLed α - 1α

1 - α/

Using Herschel ' s theorem [2, p . 73] which expresses (ez — 1) Λ a s a power

ser ies in z we obtain

00 m

α I s Q I

Comparing with

.α)= Σ βn(a)~nl

we get (3.7).

The following further properties of the numbers βn{a90L), which closely re-

semble well-known formulas for Bernoulli polynomials, are easy consequences of

the above:

( 0 < p < n ) ,

k=o

J Pn(t,CC)cit =α

lnλβk(a,a)b»-k,

Jα

n + 1

Taking a — b ~ 1 and using (3.3), we can also use this last equation to obtain

the functions βn(a, a ) recursively by successive integration of polynomials.

As a final result, taking a = 0,1,2, ,m — 1 in(3.3)and summing we obtain

U.8) Σ « = " T Γ Σ ^n+i (α. α) + — .

a generalization of the famous formula giving Σan in terms of Bernoulli poly-

nomials. This result is somewhat surprising because of the appearance of the

parameter α on the right. (When α = 1, (3.8) reduces to the Bernoulli formula.)

R E F E R E N C E S

1. N. J. Fine, Note on the Hurwitz zeta-Function, Proc. Amer. Math. S o c , scheduledto appear in vol. 2 (1951).

2. T. Fort, Finite Differences, Clarendon Press, Oxford, 1948.

oo '2kπiχ3. M. Lerch, Note sur la function ϋ(w,xts) = £ — , Acta Math. 11 (1887),

19-24. *=o (" + *>

4. R. Lipschitz, Untersuchung einer aus vier Eίementen gebildeten Reihe, J. ReineAngew. Math. 54 (1857), 313-328.

5. , Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen,J. Reine Angew. Math. 105 (1889), 127-156.

6. E. T. Whittaker and G. N. Watson, Modern Analysis, University Press, Cambridge,England, 1945.

A CHARACTERIZATION OF THE SUBGROUPS OF THE

ADDITIVE RATIONALS

Ross A. BEAUMONT AND H. S. ZUCKERMAN

1. Introduction. In the class of abelian groups every element of which (except

the identity) has infinite order, the subgroups of the additive group of rational

numbers have the simplest structure. These rational groups are the groups of rank

one, or generalized cyclic groups, an abelian group G being said to have rank one

if for any pair of elements, a ^ 0, b ^ 0, in G, there exist integers m, rc, such that

ma — nb ψ 0. Although many of the properties of these groups are known [ l ] , it

seems worthwhile to give a simple characterization from which their properties

can easily be derived. This characterization is given in Theorems 1 and 2 of §2,

and the properties of the rational groups are obtained as corollaries of these

theorems in §3. In §4, all rings which have a rational group as additive group are

determined.

Let pi, p 2 > , pi, * * be an enumeration of the primes in their natural

order; and associate with each pj an exponent kj, where kj is a nonnegative inte-

ger or the symbol °°. We consider sequences i; kγ , k2 > * * , λy, , where i is

any positive integer such that (i,py) = 1 if kj > 0, and define {i; kγ , k29* * •>

kj9 •) = (ι; kj) to be the set of all rational numbers of the form ai/b, where a

is any integer and b is an integer such that b — Ώp.pJJ with τij < kj> Then each

sequence determines a well-defined set of rational numbers. The symbol Π desig-

nates a product over an arbitrary subset of the primes that satisfy whatever condi-

tions are put on them; Π designates a product over all primes that satisfy the

given conditions.

2 Characterization of the rational groups. We show that the nontrivial sub-

groups of R are exactly the subsets (i kj) defined in the introduction.

THEOREM 1. The set (i;kj)is a subgroup o//?+, the additive group of rational

numbers. We have (i kj) — (i'; k 'j) if and only if i — i', kj = k y for all /.

Received October 11, 1950. Presente4 to the American Mathematical Society, JuneΓ7, 1950.

Pacific J. Math. 1 (1951), 169-177.

169

170 ROSS A. BEAUMONT AND H. S. ZUCKERMAN

Proof. If ai/b £ (i kj), ci/d £ (i kj), then b = H^pp, d = Π ^ . p J ^

[ i , rf] — Ώp.psJ, where SJ — max {ΠJ, mj) < kj» Writing [b, d] = bb' = dd', we

have

ai ci _ b1 ai d'ci __ (6'α ~ d1 c)i r~ ( ^ \

7 ~ 7 ~ ΐ ΰ Π "EMJ ~ ΪMΪ e U; jh

It is clear that different sequences determine different subgroups.

In the sequel we need the following properties of a subgroup G ψ 0 of R+.

(1) Every ζ £ G has the form ζ = ai/b, {ai, b) - 1, where i is the least

positive integer in G.

For every ζ we have ζ = m/b, where (m, b) = 1; and if i is the least positive

integer in G, then m = αi + r and m — ai C G imply r = 0.

(2) If αί/6 C G, ΐ C G, and (α, 6) - 1, then i/b C G.For there exist integers k, I such that ka + Zό = 1 and

(3) If ai/b C G where i is the least positive integer in G, and (α, b) = 1,

then (i, i) = 1.

By (2), t/fc C G; and if (£, 6) ^ 1, h/b' C G with A < ί. Then b' (h/b') -

h C G.

We assume in the proof of the remaining properties that the elements of G are

written in the canonical form ai/b with {ai, b) — 1 and i the least positive integer

in G.

(4) If αi/6c C G, then i/6 C G.

For cαΐ/όc = α£/6 C G and ί/i C G by (2).

(5) If ai/b, ci/d- € G, and if {b, d) = 1, then i/M £ G. For by (2) we haveFor by (2) we have

i (kb + ld)i ki li

6d 6d d b

TίfEOREM 2. If G ^ 0 is a subgroup of /?+, ί/ie i iAere exists a sequence

{i; kι , k2 , ' ' *, kj, •) such that G — {i; kj).

Proof. By (1), every ζ £ G has the form ζ = αι/6, {ai, b) = 1, where i is the

A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 171

least positive integer in G. We write all elements of G in this form. If, for every

I, there exist ai/b £ G such that p . \b, let Ay — °°. If not, let Ay — max A such

that p . I b for some ai/b £ G. Since (ai, b) = 1, we have (ί, py) = 1 if Ay > 0. By

the definition of i and kj, G is contained in (ΐ Ay). Now every element of {i; kj)

has the form ai/{pn^ p* Γ ) , where ΠJ < kj and (a, p " * p " r ) = 1. By (4) and

the definition of Ay, G contains every i/p^J with ΠJ < kj, and by(5), G contains

<u/(p?1 p? r). Hence G = (*; Ay).

3 Properties of the rational groups* In this section, properties of the rational

groups are obtained as corollaries of the theorems of § 1 .

C O R O L L A R Y 1 . The g r o u p ( i ; kj) i s a s u b g r o u p of ( i f A y ) if a n d o n l y if

j £ A yA y and i = mi'

COROLLARY 2. The group (i; kj) is cyclic if and only if kj < °° for all j and

kj = 0 for almost all j .

Proof. If (i Ay) is cyclic, it is generated by ai/b with {ai, b) = 1. Since every

element of (i Ay) has the form nai/b, we have a — 1 and 6 = Π t > 0 p .J. Con-

versely (i Ay) contains ι / Π ^ > 0 pW, and this element generates (i; kj).

COROLLARY 3. We have (£; Ay) = (*'; Ay) if and only if both kj = Ay for

almost all j , and, whenever kj ψ Ay, both are finite. Every isomorphism between

(i; kj) and (i*; Ay) is given by

αi

b

where

- = Π'

mαi

n6

πfinite

Πfinite

Proof. If (i; kj) = (i1; Ay), then i —> m ΐ ' A with (mι ;, n) = 1. If

then m7) —> mi' and ni —» mi', so that mrj = ni, or τ\ — ni/m.

Hence ni/m —» i1. We write

then for nj < kj we have

Pί\~• paJ

while for n's < Ay we have

We have the following alternatives with consequences which follow from (3):

I. j = a i : nj - k'j < α/ < kj - n'j

II. j = ^« : «) "" fe; < K 1 *> ~ Λj

III. •" ^j < fey i n'j <

It follows that kj = °° implies k'j = °° and conversely. With both Ay and A y

finite we may choose ray = Ay and n y = A y and we have:

I.

II.

III.

= fe; - fey

= fey — fey

J Φ : fey = fey

We have Ay = A y if and only if / φ Oί/, / 'φ βm. In particular, we have Ay = A y

for almost all j . If Ay > A 'y, then j = OC/ and α/ = Ay — A 'y. If A y > Ay, then j — βm

and 6 m = A y — Ay.

Now i —> mi'/n implies αι/ό —> ami'/bn, so that the only isomorphisms

between (i; Ay) and (ί 1; Ay) are those described in the corollary. Incidentally, we

A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 1 7 3

have derived necessary conditions for the relation {i kj) = W; k'j)

With the necessary conditions satisfied, we check that the given correspond-

ence actually is an isomorphism. These conditions imply that the correspondence

is single-valued with a single-valued inverse from (i; kj) onto (i1; k'j). It is clear

that addition is preserved.

COROLLARY 4. The group (i; kj)admits only the identity automorphism if and

only if kj is finite for all j .

Proof. If kj is finite for all j , we have by Corollary 3, with kj — k'j for all / ,

that m — n = 1. Conversely, if any kj — °°, then the correspondence of Corollary 3

gives us nontrivial automorphisms.

The multiplicative group of the field of rational numbers, /?*, is a direct pro-

duct of the infinite cyclic subgroups of Rx generated by the prime numbers p, for

all k. Such a subgroup consists of the elements p, , p? , , 1, 1/W > ^/p&> * * *

COROLLARY 5. The group of automorphisms of (i; kj) is isomorphic to the

direct product of all of the infinite cyclic subgroups of Rx generated by those

primes p^ for which kj = °°.

Proof. By Corollary 3, there is a (1—1) correspondence between the automor-

phisms of (i; kj) and the rational numbers U/N with (M, N) = 1, where M and N

are arbitrary products of those primes for which kj = °°. This correspondence

clearly preserves multiplication and the set of all rationale M/N has the stated

structure as a group with respect to multiplication.

COROLLARY 6. For any two subgroups (i; kj) and (£'; k'j) of /?+, the set T

consisting of all ordinary products of an element of (i kj) with an element of

(i'; k 'j) is again a subgroup of R+.

Proof. We have T = (1 Kj), where

with Sj = minίoty, k'j) + min(θty, kj), where (Xy is the highest power of pj that

divides i, and Oί'y the highest power of py that divides i'.

COROLLARY7. // ( i ; kj) > (i ;; k'j) and ph' is the maximum power of pj such

that p.i divides i '/i , then the difference group (i kj) — (i1; k'j) is a direct sum


of the groups Gj where

(i) Gj is the cyclic group,

«';*;>

Pri}

if kj is finite;

(ii) Gj is the group of type pc

(•';*}: J ΌR)

if kj is infinite and k'j is finite;

(iii) Gj — [θ] ifkj = k'j = 00,

4. Rings which have a rational group as additive group. The distributive laws

in any ring S with (i kj) as additive group are used to determine all possible

definitions of multiplication in S.

LEMMA. If S is a ring with additive group (i kj), then multiplication in S is

defined by

ai ci ac , N— X — = — 1 X 1 .6 d bd

Proof. We prove this by showing that

lai ci\ , xbd[— X— = ac (i X i) .

\b d

We have

ac (i X i) = ai X ci (by the distributive laws in S)

[ αi ail Γci ci

— + • • • + — X — + • • • + —6 6 J Id d

b summands d summands

whence ac (i X i )

[ ai ci] \ai ci] ,, , ,. ., . , . o x

— X — + + — X — (by the distributive laws in S)b d\ Lj 16 d\

bd summands

ai ciΛ

6 d

THEOREM 3. // there is an infinite number of kj such that 0 < kj < °°, then

the only ring S with (i kj) as additive group is the null ring. If 0 < kj < °° for

only a finite number of kj, then S is a ring with additive group (i kj) if and only

if multiplication in S is defined by

bd Π' pp

where A ' and nj are arbitrary.

Proof. If S is a ring with additive group (i kj), then i X i = Ai/B £ (i;kj)9

where (Ai, B) = 1, B = Π p W, nj < kj. By the lemma, we have

ai ci acAί

b d ~ bdB

If 0 < kr < °°, this yields in particular

i i Ai

Therefore (pΓ, B) = 1, for otherwise we would have 2kr + nr < kΓ9 which is im-

possible. Hence, B = Π p?J is a product of primes for which kj = °°, and it is

necessary that p^ΓU. If there is an infinite number of primes py with 0 < kj < °°,

then A — 0 and (ai/b) X (ci/d) = O This proves the first statement in the theorem.

If 0 < kj < °° for only a finite number of primes p. , then

A = A' Π p)i

176 ROSS A.BEAUMONT AND H. S. ZUCKERMAN

Together with what has been proved above, this gives

acA' j Π Pjai ci _ \o<kj<co

X — • ,

b d _ , '

bd Π P?}

j

where A ' and nj > 0 are arbitrary integers.

Conversely, this definition of multiplication always makes (i; kj) a ring. Closure

with respect to X is insured by providing p .3 in the numerator when 0 < kj <°°,

and the associative and distributive laws are readily verified.

COROLLARY l The set {i; kj) is a subring of R if and only if there is no

kj such that 0 < kj < °°.

Proof. Let (i kj) be a subring of R and assume that for at least one kj we

have 0 < kj < °°. If 0 < kj < °° for infinitely many kj, then (ί kj) is not a subring

of R, since by Theorem 3 it is the null ring. If 0 < ky < °° for a finite number of

kj, then multiplication in any ring with (i kj) as additive group is given by the

formula of the theorem. Hence this must reduce to ordinary multiplication for some

choice of A ' and nj that is,

A' Π r J

Π'pW= i; A' Π p / ' = i Π ' P ; ι .

By hypothesis, at least one pj with kj > 0 appears in the left member of the above

equality. Since no prime appears in both products, we have pj\i. This contradicts

(i, pj) = 1 for kj > 0.

Conversely, let every kj be either 0 or oo, By the theorem, we have

a t

6X

Cl

dbd

acΛ i

rr γ

and we may select A1 = i, Π ^.=00 pni ~ 1, yielding ordinary multiplication.

C O R O L L A R Y 2 . // ( i ; kj) is a subring of R , then ( i ; kj) is a ring under the


multiplication

ax ci ac lei

b d ~ bd \ f

for arbitrary ei/f £ {i kj).

Proof. By Corollary 1, we have kj ~ 0 or kj = °°, so that every element of

(i kj) has the form

A'i

ir $*

and by the theorem these are just the multipliers which are used to define multi-

plication.

COROLLARY 3. If S is a ring with additive group {i; kj), then either S is a

null ring or S is isomorphic to a subring of R,

Proof. If S is not null, the correspondence

ai aA ai ci acAi— —> — - , where — X — = ,6 bB b d bdB

is (1~1) from S on a subset of R, and

ai ci (da + bc)i (da + bc)A __ aA cA

b d bd bdB bB dB '

ai ci acAi acA2 __ aA cA

b d ~ bdB bdB2 ~ bB dB

COROLLARY 4. All rings with additive group Λ+ are isomorphic to R.

Proof. The correspondence of Corollary 3 clearly exhausts R.

REFERENCE

1. R. Baer, Abelian groups without elemeμts of finite order. Duke Math. J. 3 (1937),68-122.

U N I V E R S I T Y O F WASHINGTON

RECURRENCE TIMES FOR THE EHRENFEST MODEL

RICHARD BELLMAN AND T H E O D O R E HARRIS

1. Introduction and summary. In 1907, P. and T. Ehrenfest [ l ] used a simple

urn scheme as a pedagogic device to elucidate some apparent paradoxes in thermo-

dynamic theory. Their model undergoes fluctuations intuitively related to fluctu-

ations about equilibrium of certain thermodynamic systems. In view of an apparent

discord among physicists [6, pp. 139-145] we shall not try to force an analogy

with entropy.

The original Ehrenfest scheme was defined as follows. Initially, 2/V balls are

divided in an arbitrary manner between two urns, 1 and 2, the balls being numbered

from 1 to 2N An integer between 1 and 27V is selected at random, each such

integer having probability (2/V)~1, and the ball with the number selected is trans-

ferred from one urn to the other. The process is repeated any number of times. If

πγ and n2 are the numbers of balls in urns 1 and 2 respectively before a transfer,

it is clear that the probability is nχ/(2N) that the transfer is from urn 1 to urn 2

and n2/(2N) that it is in the contrary direction.

Let x' (n) be the number of balls in urn 1 after n transfers, and let L 'j9k be

be the smallest integer m such that x' (m) = k9 given that x' (0) == /. If k = /, we

call L'k9k the recurrence time for the state k. If k ψ j , we call L 'j^ the first*

passage time from j to k The distribution of %'(n), known classically, was

derived by Kac [5] as an example of the use of matrix methods. Kac then found

the mean and variance of £/,&> attributing some of his methods to Uhlenbeck

Friedman [4] found the moment-generating function for x1 (n) (for the Ehrenfest

and more general models) by solving a difference-differential equation.

Instead of the original Ehrenfest model, we shall discuss a modified scheme

with a continuous time parameter, which was apparently first suggested by

A. J.F.Siegert [ 9 ] . In this scheme there are two urns and 2N balls initially

divided between them arbitrarily. Each ball acts, independently of all the others,

as follows: there is a probability of (1/2) it + o(dt) that the ball changes urns be-

tween t and t + it, and a probability of 1 — [(1/2)it + o(it)] that the ball remains

Received August 5, 1950, and, in revised form, November 8, 1950.Pacific J. Math. 1 (1951), 179-193.

179

1 8 0 RICHARD BELLMAN AND THEODORE HARRIS

in place between t and t + dt. Standard reasoning then shows that the total proba-

bility of a change by some ball between t and t + dt is Ndt + o(dt), and that

consequently the probability density for the length of time between transfers is

Ne l dt. When a transfer occurs, it is readily seen that the probabilities that it

is from urn 1 to urn 2 or from urn 2 to urn 1, respectively, depend on the relative

number of balls in the two urns exactly as for the original Ehrenfest model. Thus

we see that the present scheme is essentially the original Ehrenfest scheme where

the drawings are made at random times. As we shall see, the time-continuous

scheme is easier to handle analytically.

Let x{t) be the number of balls in urn 1 at time t we shall sometimes speak of

this number as the state of the system. Then x(t) is a random function which can

take integer values from 0 to 2/V; x(t) executes a random walk—with a "restoring

force7-—about the equilibrium value N. It is clear that the random walk is a Markov

process.

Let Lj9k, j f1 k, be the first-passage time from state / to state k that is, Lj9k

is the' infimum of t such that x{t) — k, given that x(0) = /• Let L^^ be the re-

currence time for the state k; that is, L^^ is the infimum of t such that x(t) = k

and x(r) φ h for 0 < r < ί, given that x(0) = k. We shall discuss the probability

distributions of L; jς and L^^

The probability distribution of Lj^ depends, of course, on the size of the

model (that is, on the number N). When it is necessary to emphasize this de-

pendence we shall sometimes employ the notation L- £ in place of £/,&•

We shall use the notation P{A) for the probability of the event A P(A \ B) for

the conditional probability of A, given B E(X) for the mean, or expected value, of

the random variable X. By the distribution of a random variable X we mean the

function (of say u) given by P(X < u). The statement that a sequence of dis-

tributions converges to a distribution F(u) will mean convergence at all continuity

points of F(u).

There are two limiting situations in which the distribution of £/,£ is of interest.

(a) Consider a simple thermodynamic system such as an ideal gas in a con-

tainer. Let us think of the container as consisting of two halves which, however,

are not separated by a partition. Suppose that initially the molecules are spread

in a rather uniform manner through the two halves of the container. According to

classical kinetic theory, if we wait long enough, a time will come, in general,

when all the molecules are in one half of the container. Such events, where the

fraction of molecules in one half of the container is appreciably different from

RECURRENCE TIMES FOR THE EHRENFEST MODEL 1 8 1

(1/2), are evidently enormously rare if the number of molecules is large. Corre-

spondingly, we should like to show that the random variable £#,& > where | k ~~ N \

is of the order of magnitude of N, is very large with high probability when N is

large. Now the mean of L#tk 1S extremely large when N is large. However, as Kac

has observed, the standard deviation is of the same order of magnitude as the

mean. Thus we cannot conclude from the values of the first two moments that L # ^

is large with high probability. We shall show, however, that the distribution of

Ltf 9yE{L^fj£) converges to 1 ~~ e u as N—> °° provided k/N remains less than

some fixed number λ t < 1 (Theorem 1).

The situation with respect to L/ς £, where again k/N < λj < 1, is somewhat

different. If k/N is appreciably different from 0, a very short recurrence time is

not improbable. The distribution of L^^/EiL^^) has for large N a "lump"of

probability of magnitude k/N concentrated near 0, the remainder of the distribution

being exponential (Theorem 2).

(b) In the theory of the Brownian motion and elsewhere in physics and sta-

tistics an important role is played by the stationary Gaussian Markov process z(t)

which we scale so that

E[z(t)] = 0, E[z{t)Y = 1/2.

This process is defined bythe requirement that the joint distribution of z(tι), ,

z(tm) for any distinct numbers t\, , £m is Gaussian and dependent only on the

differences t( ~~tj and that the autocorrelation function is given by

If N is large, the z{t) process, under the conditional hypothesis that z(0) has an

appropriate value, is approximated by the process

x(t) -N

in a sense described in Section 6. (It should be remembered that x(t) depends on

N.) By considering the distribution of Lpt^ where (k — N)/NΪ/2 —» — ξQ < 0,

/Y —> oo We obtain in Theorem 3 the Laplace transform of the distribution of L,

the first time at which z(t) = — ξOy given z(0) = 0. This result is not new, having

been obtained by Siegert [lOj and by Darling (unpublished). However, the present

method of derivation seems instructive.

Results similar to those given under (a) and (b) are obtained for the random


variable L%9k , the first time | x(t) — TV | = N ~~ k9 given #(0) = N,

2 The mathematical model* Suppose that there are initially j balls in urn land

2/V — / in urn 2. Associate with the ith ball a random function x((t) defined as

follows: Xi(t) is 1 if the ith ball is in urn 1 at time ί, and 0 otherwise. From the

elementary theory of Markov processes (see, for example, Kolmogorov [7] ) , we

have

2

We may define the generating function of *, (i) by

P[*i(t)=0] +-sP[xi(t)=l]

Then the generating function of xjit) is, from (1),

or

according as *j(0) is 0 or 1. Since the quantities Λ, (ί), i = 1, * , 2/V, are inde-

pendent, the generating function for x(t) — Σ Xi(t) is2N

(2) Σ Pί*(t) = k\x(0) = j]sk

k=o

= 2"2Λf[l ~ e-f + (1 + e-*)sV[l + e'* + (l - β" ' )* ] 2 *^

= ΣQj,kU)sk,k = 0

where we have introduced the notation Qjfk(t) for P [x(t) = A; | #(0) = y] . Formula

(2) was given by Siegert [9 111

Because of the simple nature of the process under consideration it is easy to

show that Lj9k and L^^ are (measurable) random variables with absolutely con-

tinuous distributions. We omit the proof. We let Py^(α) be the probability density

Lj,k>

S0

UPj,k(y)dy=P[Ljfk<u].

Define

(5) mk = l/[(N-k)Qk], k<N.

It is convenient to notice that, as N—> °°,

where we have put λ •= k/N, and OiX/N) is independent of λ .

The quantities Ljfk> (?ifc> a n <^ s o o n> depend on the size of the model; when it

is necessary to emphasize this dependence we shall write L-^fQ^ , and so on.

3 Distribution of £/,&, / ψ k. In this section we consider the distribution of

the first-passage time from state / to state k for large N, where | / — k \ is of the

order of magnitude of /V. As far as the limiting distributions are concerned, we can

restrict ourselves without loss of generality to consideration of LΛ',^> k < N. For

example, if j > N > k then we can write

Lj,k ~Lj,N +£tf,fe

The first-passage time from j to N, representing movement toward equilibrium, is

negligible relative to Lsfk a n <^ d ° e s n o t affect the asymptotic result. On the other

hand if N > / > A, we have

and it is not difficult to show that Ljqj is negligible compared with Ln9fg .

If the first passage to the state k occurs at time r , the probability that the

state at time t is again k is Qk,k^ "" T ) . We have therefore

(7) Qjlk(t) = S*Pj,k (r)Qk,k (t-r) dτ, j φk.

Formula (7) is the continuous counterpart of a formula long used for discrete

184 RICHARD BELLMAN AND THEODORE HARRIS

processes and recently exploited by Feller [2] . Taking Laplace transforms of

both sides of (7) we have

S0Qj.k()( 0 e'σt dt = , R(σ) > 0

Γ

Since the quantities Qjfk^ a r e polynomials in e **, as we observe from (2), both

the numerator and the denominator in (8) have a simple pole at σ — 0, and their

quotient is therefore analytic in the circle \σ\ < 1.

For simplicity denote LJ^ ^ by LJj. We have the following result.

THEOREM 1. The distribution function of L^ /m^ ' converges to 1 — e ",

\L > 0, as N—> °°, provided k/N < λ t < 1, the convergence being uniform in k

and u,

The proof will bring out the fact that likewise

(9) E[L[N)]/m^ —» 1 , N—>&, k/N < A x .

Theorem 1 will follow from this lemma:

LEMMA 1. For the complex variable σ, let

Further, let \k(N)] be a sequence of nonnegative integers such that k(N)/N —» λ0

< Xι < 1 as N—> °°. Then (the convergence being bounded and uniform provided

|cr| <σ0 < 1),

(10) lmΦiyi(σ)=- , H < 1 .

Proof of Theorem 1. The function φ^. '( — σ) is the moment-generating

function" of the quantity L^, ymj. . Lemma 1 then implies, as is well known, that

RECURRENCE TIMES FOR THE EHRENFEST MODEL 185

uniformly for u > 0 provided k(N)/N —» λ 0 . Lemma 1 also implies, since we have

convergence in a complex neighborhood of σ = 0, that

so that (11) is still true if we replace m j ^ by E J ^ )

Now if Theorem 1 were not true then an € > 0 and a sequence \h(N)\ ,h(N)/N

< λ t , would exist such that for infinitely many integers N we would have

Extracting a convergent subsequence from \h{N)/N], we are led to a contradiction

of (11).

Proof of Lemma 1. The proof of Lemma 1, which is somewhat indirect, pro-

ceeds as follows. We can obtain an expression for φ\ {&) by substituting cr/m\

for σ in (8), obtaining

(12) φίM) (σ) =T

-j

We can obtain an asymptotic estimate of J γ as we shall see later. However, a di-

rect estimate of J2 appears difficult to obtain. We shall therefore resort to another

expression for φ^ (σ) which is easier to estimate. Having estimates for φ^ '(σ)

and for / ι , we can get an estimate of / 2 , which will be necessary for Theorem 2.

Since a direct proof of Lemma 1 is easy if all terms in the sequence {k(N)\ are

0 we can suppose k > 0. If 0 < k < N we have, from elementary reasoning, the

important relation

On account of the Markovian nature of the process, Ltf^ and L^> 0

a Γ e independent

random variables and the Laplace transform of the distribution of their sum is the

product of the Laplace transforms of their individual distributions. Therefore,

using (8) and (13), we have

(14) E(e~sLN>° ) =E(e~slN>k ) £ ( e ~ s L M ) ,

OΓ

(15) φ[NHσ)= Jβ

βflr,* ( 0 .-**/•**

/o°° PΛ.O (t) Γ*" * dt j Γ ft,, (t) e-'/-* *

Γ Pfe,o (t) e-σt/ * (ft Γ ft,, (t) e-'/ * dt

The advantage of (15) over (12) is that Qk,o(t) is a simpler function than

The numerator of the last fraction in (15) is (l/2)β [N + 1, (l/2)σ/m^ 3 . The

denominator, with the substitution e t = y , becomes

^XOj X "" I ( 1 """" Y ) ( 1 "f" V ) y>CV»'*R' *• fj y ^

We now have to estimate I as N —> °° under the hypothesis k/N—> λ 0 < 1.

[We shall write simply k for k(N).] We shall restrict σ to the circumference of a

circle, say \σ\ = (1/2), since it is clearly sufficient to prove Lemma JL for such

a circle. Write

1 = f6 + f1 =I i + I 2 , 0 < € < l - λ o < l .

Making use of (6) and the fact that (1 — y)k (1 + y)2 N "k increases to a maximum

at y = 1 — k/N and then decreases, 1 — k/N being larger than € for sufficiently

large N9 we have

/T7\ T Z l/iV2 , /•€

L-y)fe(i+:

= ~ [l + o(l)] +O[(l-e) f e (l +e)2J™ logJV]σ

= — [l+o(l)j +o(mk),σ

where o( ) is independent of σ for \σ\ = 1/2.

To estimate I2 we distinguish the cases \ 0 > 0 and λ 0 = 0. If λ 0 > 0, then

12 can be estimated using the method of Laplace; see [8, p. 77] . We obtain then,

setting k/N = λ, (see (6)),

(18)ττλ(2 ~ λ) 1/2

ΛT(l-λ)

Going back to (15), we obtain (10) from (17) and (18), since

(1/2)B[N + 1, (l/2)σ/mk] = (**/er)[l + <

This completes the proof of Lemma 1 for the case λ 0 > 0. If λ 0 = 0, the integral

12 can be estimated by making a change of the variable of integration which shows

the integral to be asymptotically equivalent to a Beta function. We need not enter

into details.

4. Distribution of Lk9k We shall establish the following result.

THEOREM 2. Assume λ = k/N < λt < 1, and put

Fλ(u) = λ + (1 - λ) [1 - e~( h λ ) u ] , u > 0 .

Then for every b > 0 we have

lim ^ K4^O^- f λ( u ) l = 0

uniformly in k.

Proof. As in the case of Theorem 1, it is sufficient to prove that the Laplace

transform of the distribution of NQ£ L5 ? approaches

λ0 +(l"-λo)V( o)

provided k/N = λ —> λ 0 < 1. (We know from the general theory of Markov proc-

esses that


see Feller, [3, p. 325].)

The relation which replaces (7) when / = k is

(19) Qk,k (t)=e'Nt + JΓ* Pk>k (r)Qkιk [t -r) dr ,

the term e t in (19) being the probability that the system remains in state k the

entire time from 0 to t. From (19), we have

(20) / ^We

If we equate the right side of (8), with j — N and with σ replaced by crNQk, to

the rignt side of (15) with σ/τn^ replaced by σNQk, we obtain

or

(21) JΓ Qk.k(t)

To estimate /3 , which is the numerator of (12) with σ replaced by σ*/(l — λ), we

need two lemmas.

LEMMA 2. Given e > 0, let

t: max

Then

(22) tff(e)=O(logJV), N—>™.

Proof. By (2), QNtΓ(t) is the coefficient of sΓ in

where z — z(t) = 2(1 + e~2t)/(l — e~2t). Since for large N the root of the

equation

(1 - e~2t)N = 1 - δ , 0 < δ < 1 f δ fixed ,

is approximately t ~ (1/2) log Nf it suffices to prove Lemma 2 for the quantities

\cr ( t) — cr \ Itjy(e) = sup ' t : max jj^\ > er ,

- r J

where we have set

(23) ( 1 + 2 S + s 2 ) " :

C C ( ) 2 Q

Suppose e > 0 is given.Choose an arbitrary OC > 1. Let 6ι < e be a positive

number and define

(24) e f f + 1 = e w ( l + W α ) , N = l,2, " .

Note that { βff \ is a bounded increasing sequence. We select €x small enough so

that eN < € , for all N. Now define a sequence ί\ < T2 < as follows:

Fx = tι(βι) ί^+1 for yV > 1 is the maximum of TN and the positive root of

(25) (l/3)[Z(t) -2](l+eH)/eκ = 1/N«.

[Note that zit) is monotone decreasing.] It is then clear from (25) that

(26) tN - (l/2)α log N, N—>oo.

We now wish to show inductively that

icΓ

wω-cW|(27) ^ j <eN f o r t>tN, N = l , 2 , ' .

Clearly (27) holds for N = 1, since βι < e and7 t = t\ Suppose that (27) is true

for a general N From (23), we have

(28) c<»+1> (0 = e<*> (t) + zc\i\{t) + c ί 5 ( 0 ,

c^=cW+2c\l\+c[ί\.

Using (27), (28), and the fact that < % / < > / * + l ) < 1/3, we have for t > ΊH ,

From the definition of €# it is then clear that (27) holds with N replaced by N + 1.

Then t > 7/v implies that the left side of (27) is less than € . Use of (26) now com-

pletes the proof of Lemma 2.

LEMMA 3. Assume k/N < λι. Then

Q J 5 (0 < exp[ ~3(1 - λi)2iV/5] .β

Proof. Lemma 3 is an immediate consequence of a result of S. Bernstein on

sums of independent random variables; see Uspensky [12, p. 205] . To apply

Bernstein's result, we consider the 2N balls as consisting of N pairs, each pair

having initially one ball in urn 1 and one in urn 2, letting Uspensky's random vari-

variable x t be the number of balls from the iih pair in urn 1, minus 1, at time L

Now

N

-p Σ * * = * - * <p\Σ*i<k-N\,

and the applicability of Bernstein's result is obvious.

We now return to the proof of Theorem 2. To estimate the integral /3 defined

in (21), write

( 3 0 ) h= S0

C°(QN,k(t)-Qk)e-σNQktdt +

Write the integral on the right side of (30) as


for an arbitrary € > 0, where ίjy(e) is defined in Lemma 2. Using Lemmas 2 and

3, we have

(31)

\Γ3 I =0{ίog N exp[-3(l - λtfN/5]} .

Thus /3 ~ l/uVcr) Putting this estimate in (21) and recalling from Theorem 1 that

i+°/(i-λ0) '

we get the desired result from (20).

5 Intuitive interpretation. Theorem 1 means intuitively that if we take m^

as our time unit, the attainment of the state k is an occurrence of the "chance"

type; that is, the probability of attaining k during a given time interval is almost

independent of the past history of the process. This interpretation suggests that

Theorem 1 should be true for more general types of processes with a central

tendency.

Theorem 2 seems to mean that if the initial state is k there is a probability λ

of returning J o k before leaving its immediate neighborhood; there is a probability

1 — λ of getting completely away from the neighborhood before the first return; in

this case the first return has the distribution of first passage times given in Theo-

rem 1.

6* Application to stationary Gaussian Markov processes* In Theorems 1 and

2 we considered rare or microscopic fluctuations of x(t). But if N is large x(t) will

for the most part deviate little from its mean value /V, and to consider the ordinary

fluctuations of x(t) we consider

Let ί ι , , t m b e a fixed set of nonnegative numbers. The joint distribution of

Ztf(tι)* * * * 9 Ztfίtm)* given 2^(0) = 0, approaches, as N—* °°, the joint distrib-

ution of z(tι)f , z(tm), given z(0) = 0, where z(t) is the stationary Gaussian

Markov process with

£ [ Z ( t ) ] = 0 , E[z(s) z(s + t)] = (1/2) e " " 1 .

Define the random variable L to be the smallest value of t for which z(t) =

~~£o ^ 0> given z(0) = 0. It is intuitively clear that the distribution of L is given

by the limiting distribution of L^9k a s N—* °° provided we let

(32) (fc-jvj^i/a—* _ £ , .

A rigorous proof of this statement is not difficult but we omit it.

To find the limiting Laplace transform for the distribution of Ljv,& u nder the

hypothesis (32), we consider (15) with σ > 0 in place of σ/m^ , and let k =

N — ξN^2 . The substitution e t — y/N^2 puts the denominator in the form

\

7Nl/2j

where Ct is an arbitrary number between 0 and 1/6 . If 0 < y < /Vα, then

Hence,

N1/2I

mξuιn

+ θ(ΛΓ 1 / 2 + 3 α ) ] .

yσ l dy.

The second integral inside the bracket in (33) goes to 0 as /V —> °° .The numerator of (15), with σ in place of σ/m&, is

We thus have the following result.

THEOREM 3. The Laplace transform of the distribution of L is given by

(l/2)Γ(σ/2)(34)

Formula (34) was obtained by Siegert and by Darling through direct consider-

ation of the z(t) process. It is interesting to notice that the present procedure

utilizes (13) which has no counterpart for the z(t) process.


7. Two-sided limits. Let L%\k9 N > k, be the first time | x (ί) - N | = N ~ k9

given x(0) = N. Let L* be the first time \z(t)\ = ξQ > 0, given z(0) = 0. Argu-

ments similar to those used for Theorems 1 and 3 give the following two results,

THEOREM la. Under the conditions of Theorem 1 the limiting distribution of

L%9k/mk is 1 ~~ e ~ 2 u , a > 0.

THEOREM 3a. The distribution of L has the Laplace transform

(l/2)Γ(σ/2)

f e y γσ ι cosh (2ξ0y)dy

8. Added in proof An argument has been found which rigorizes the remarks of

Section 5 and gives a proof of Theorems 1 and 2 for more general processes.

REFERENCES

1. P. and T. Ehrenfest, JJber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem, Phys. Z. 8 (1907), 311.

2. W. Feller, Fluctuation theory of recurrent events, Trans. Amer. Math. Soc. 67 (1949),98-119.

3. , Introduction to Probability, Wifey, New York, 1950.

4. Bernard Friedman, A simple urn model, Comm. Pure Appl. Math. 11 (1949), 59.

5. Mark Kac, Random walk and the theory of Brownian motion, Amer. Math.- Monthly 54(1947), 369.

6. A. I. Khinchin, Statistical Mechanics, Dover, New York, 1949, pp. 139-145.

7. A. Kolmogorov, Analytischen methoden in der Wahrscheinlichkeitsrechnung, Math.Ann. 104 (1931), 432.

8. G. Polya and G. Szegό', Aufgaben u, Lehrs'άtze I, Dover, New York, 1945.

9. A. J. F. Siegert, Note on the Ehrenfest problem, Los Alamos Scientific Laboratory,MDDC-1406 (LADC-438).

10. , On the first passage time problem (abstract)t Physical Rev. 70 (1946),449.

11. , On the approach to statistical equilibrium, Physical Rev. 76 (1949),1708-1714.

12. J. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill, New York,1937.

STANFORD UNIVERSITY

THE RAND CORPORATION

ON THE APPROXIMATION OF A FUNCTION OF SEVERALVARIABLES BY THE SUM OF FUNCTIONS

OF FEWER VARIABLES

S. P . D I L I B E R T O AND E . G . S T R A U S

1. The problems. Let R denote the unit square 0<x<lf0<y<l9 and

C R the space of all continuous real-valued functions z defined on /?, with norm

11 z 11 defined by | | z | | = ma.x(X9y)€R \z\. Let Ix and Iy denote respectively

the unit intervals 0 < x < 1 and 0 < y < 1; and let Cx and Cy denote respective-

ly the classes of all continuous functions on Ix and ly By an obvious identifi-

cation Cx and Cy may be considered as subsets of C R Let C$ denote the subset

of C R composed of all functions z £ CJJ such that z = / -f g where / £ Cx and

g C Cy C$ is closed (under the above norm)

For z £ CR , define the functional^/i [ z ] by

μ[z] — d i s t [z, Cs] ~ inf | | z — u; | | .

The following problem was posed by The RAND Corporation.

Problem (A): Given z £ C/j and e > 0, give a method for evaluating μ[z~\ to

within € . *

Problem (B): Given z £ C/j and 6 > 0, give a method for constructing

functions / £ C^ and g £ Cy such that

It is our purpose in the present note to solve these problems and to establish

certain generalizations.

Received November 7, 1950.

•Actually, this differs somewhat from the problem as formulated by RAND, which was:Given z and 8, give a method for determining whether μ-[z] < δ This is in all probabilityunsolvable when μ[z] = δ, since any computation of μ[z~\ which can be carried out in afinite number of steps will, in general, yield only an approximation.

Pacific /. Math. 1 (1951), 195-210.

195

196 S. P. DILIBERTO AND E.G.STRAUS

2. The role of the minimizing sequence. We shall now define a few terms by

means of which our procedure can be outlined conveniently.

We shall say that two functions z and z in CR are equivalent if z —z £ Cs,

and shall denote the equivalence of z and z by z ~ z . Clearly, z ~ z implies

μlz] = μ[z]According to the definition of μ [ z ] , there exists a sequence of functions

\wι\, W{ C C$1 such that

v>ί\

Let us define Z( — z — wι; then z^ ~ z and ||2:^|| —> μ [z ] . We shall call a

sequence fzj }, z/ C C/? , a minimizing sequence for z if z t ~ z for all i and

Clearly, both of the proposed problems will be solved once one has constructed

a minimizing sequence.*

We shall introduce a "leveling process," which when applied to z and then

iterated will produce a sequence of functions \zι\ with the properties (1) Z( ~ z

and (2) | |z, || > | |z, + i | | for all i. Properties (1) and (2) imply

lim || zi | | = if > μ[z] .i-oo

That M = yU [z ] , that is, that our "leveling sequence" is in fact a minimizing

sequence, is the principal result of this paper.

This will be established by a "pincers" argument to obtain convergence—

μ [ z j is simultaneously approximated from above and below: For each path in the

class of admissible paths L (defined below) we shall define a functional 77j[z],

over Cβ , with the important property 77 [z ] < μ [z ] . Let

S U P I77/ [ z J I = τf[.z~\leL

Then

-rr[z] < μ [ z ] < M .

* Given a sequence of real numbers ai~^at let us call the integer-valued function N(e)of the real variable € , defined for € ^ 0 , a modulus of convergence for the sequence a j , ifi £ N(ε) implies | α i - σ | < c

While a method for constructing a minimizing sequence answers the questions, thefiniteness of the procedure is satisfactory only when one has an estimate for the modulusof convergence. This will be discussed at the end of this paper.

APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 1 9 7

Our proof is accomplished by showing that 77[_z ] = M, thus implying also that

ττ[z] = μ[z] = M.

3 The main theorem. We shall say that a closed polygonal line is permissible

if it lies entirely within the square 0 < x < 1, 0 < y < 1, and if each of its

sides is parallel either to the x- or to the y-axis.

We enumerate the vertices of a permissible line by (xj>yj)9 j — 1,2, ,

where

l - * 2 f e ι y2kJtl~y2ky *fe + 2n = *fe, ϊk + 2n = ϊk \ k = 1, 2, ' .

To each permissible polygonal line I we can associate a functional π^ \_z~\ with

LEMMA 1. If z ~ z, then ^ [ 2 ] = rr^z].

Proof. Let z(x,y) = z(x,y) + gθc) + h(y) then

2n 2n 2n 2π

Σ(-i);>U;,yy) = Σ(-i)^(*j,yy) + Σ(-i)'β(*j) + Σ>=1 j-ί j-l > = 1

But

Σ (-l)yg(*i) = - Σ g(*Λ-i) + Σ «(*2k) = 0 ,

2n

Hence

2n 2n

Σ (-i)''*(*j.yj)= Σ

that is, 77j[>] = 77 z[z].

We remark that these invariants (under equivalence) ^ [ 2 ] form a complete set

of invariants. That is to say: If 77, [z ] = 77, [2] for all permissible lines, then

198 S. P. DILIBERTO AND E. G. STRAUS

z ~ z . In fact the 77, based on rectangles alone form a complete set of invariants.

In order to relate μ with the 77 we prove the following result.

LEMMA 2. The functional μ [ z ] satisfies μ [ z ] > | 77 [ z ] | for all permissi-

ble lines.

Proof. If we had μ [ z ] < 77 [ z ] | then there would exist a function z ~ z

such that I z || < I 77 [ z ] | and hence :

λ 2n

I w J I - \"il*s\ - 2n

•i 2 n

J = l

which is a contradiction.

Problem A will be solved once we establ ish the following theorem.

THEOREM 1. The functional μ\_z~\ satisfies μ [ z ] = s u p | π ^ [ z ] | = 7 7 [ z ] ,

where the sup is taken over all permissible lines.

As a preliminary to the proof of this theorem we introduce the following level-

ing process :

Given z £ CR , we define the sequences of functions zn(x,y), gn(χ)i AΛ(y)

(n — 1, 2, ) by the relations :

Z i =z Z t Z2n ~ Z 2n-1 βn > Z2n+1 Z2n ^ π >

min

It is obvious that zn ~ z(n = 1,2, ) The passage from z2n~ι t o Z2n reduces

liz2Λ-ilι ^y t-ne maximal amount by which it can be reduced through the sub-

traction of a function of x , while the passage from z2n t° Z 2n+i reduces ||z2nll

by the maximal amount by which it can be reduced through the subtraction of a

function of y. Thus, if we let

= \\z

APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 199

then the Mn form a nonincreasing sequence of nonnegative numbers, so that

limn^a)Mn[^ z ] — Aί [ z ] exists. We have the following obvious result.

L E M M A 3 . The f u n c t i o n a l μ [ z ] s a t i s f i e s μ [ z ] < M [ z ] .

Our solution of problem B will be a consequence of the following theorem.

THEOREM 2. The functional μ [ z ] satisfies μ[z~\ = /W [ z ] .

This, incidentally, will establish the fact that the functional Aί[z] is invariant

under equivalence. The direct proof of this fact might prove somewhat cumbersome.

Keeping in mind the results of Lemmas 2 and 3 we see that both Theorems 1

and 2 are consequences of the following result.

THEOREM 3. The functional π f z ] satisfies ττ[_z~\ = Λ f [ z J .

Proof. We shall call a function z horizontally level if

max z(x,y) = — min z(x,y)

for 0 < y < 1, and we shall call it vertically level if

max z{x,y) = — min z{x,y)

for 0 < x < l.For the sake of brevity we shall use the symbol M instead of Λ/[zJ.

There exists a number N such that M2N < M + €, where e is a small positive

number which is to be further determined later. We now perform the next 2τz steps

of the leveling process on the function z2N

There exists a point (xι9yχ) such that

and since z2N+2n i s vertically level there exists a point (^2,72) with *2

such that

Hence we have

200 S. P . DILIBERTO AND E. G. STRAUS

and since M2N + 2n-ι < M + € and Xι — x2 this implies

M + e-gN+n(Xl)>M + 8 or g ^ + n ( χ 1 ) < e - δ

- M - e - gN+n(Xl) <-M-δ or gN+n(Xl) > δ - e

We therefore have certainly

~ e < g/v + n (^ i ) < e

Thus

and since z2jv + 2rc-i ^ s n o r i z o n t ; a H y level there exis ts a point ( # 3 , 7 3 ) with

} s = 72 s u c n that

By the same process a s we applied to gjV+rc(#i) w e c a n now show that

_1(y2) <2e

hence

fur ther, b e c a u s e z2iV + 2rc-2 i s v e r t i c a l l y l e v e l , t h e r e e x i s t s a p o i n t (χ4,y4) w i th

x4 = %3 s u c h t h a t

R e p e a t i n g t h i s p r o c e s s 2n t i m e s w e f ina l ly o b t a i n a s e q u e n c e of p o i n t s ( % i , y i ) ,

' # > ( * 2 Λ + 1 » 7 2 7 1 + l ) j s u c n t n a t ^ 2 / c ~ ^ 2 / c - l ' 7 2 ^ + 1 = Ύ 2 k ( * = 1 , * # f » )

and

z2N(x2k-vy2k-J >M + S-(22n~1 - l )e (fe = l, ,π + 1 ) ,

z2N(x2k y2k) < - M ~ δ + ( 2 2 n " 1 - l ) e (fe = l, , n ) .

We complete the above sequence of points to form a permissible line by adding

the point U 2 n + 2> y2n^2^ w i t n *2rc + 2 = *2rc+l » 72/1 + 2 = 7ι I f w e construct thefunctional 77 associated with this permissible line then we obtain

h M I = \ττι[z2N\\ = —1 2Π + 2

> It + δ - (2 2 *" 1 - l)e — (M + e) .

Since the choice of 6 was independent of n, we can choose e so that (22n~ι)e

= 6x/2 where e% is an arbitrary small positive number. At the same time we can

choose n so large that

M + € €!

n + 1 2 '

Thus we have: For every βγ > 0 there exists a permissible line such that

or, in other words,

τ r [ z ] > M [ 2 ] .

In conjunction with Lemmas 2 and 3, this proves Theorem 3.

4» The discontinuous case. Examining our method of proof we can make the

following observations:

(1) No essential use was made of the continuity of any of the functions

z(x9y)i g(χ) > h(y) involved in the definition of μ [ z ] Specifically we may define

μ*[z] = infg,h supo<;*£ifo<cy<i I z(x9y) - g(x) ~ h(y) \ ,

where z is an arbitrary (bounded) function defined for 0 < # < I, 0 < y < 1 and

g(x), h(y) are arbitrary functions defined over 0 < x < 1 and 0 < y < 1, re-

spectively. The definition of 77[z] remains valid for discontinuous z , while Λ/[zJ

can be extended to a functional M * [z ] which is defined for discontinuous (bound-

ed) z, simply by replacing all the max and min symbols in the leveling process by

sup and inf symbols respectively. With very minor modifications of the proof of

Theorem 3 we then obtain the following result.

THEOREM 3*. The functional rr[z] satisfies π[z] = μ * [ z ] = M*[z] ,

where (unless we wish to allow infinite values for these functionals) z is an

arbitrary bounded function.

Theorems 3 and 3* yield the following corollary.

COROLLARY. If z is continuous, then

In other words9 the approximation of a continuous z(x,y) cannot be improved by

permitting discontinuous g(x) + h(y).

(2) The functions 77, [z] are continuous functionals in our metric; more spe-

cifically, we have the following result.

LEMMA 4* // \\z — z\\ < e , then | ^

line.

Proof. We have

< e for any permissible

2n

Hence

2 n

2 n j

2n

Σ (-i)'l>(*/.yj) " *(*>.y>:

1 2n

<— Σ2π > = χ

1ϊ II < — 2ne=e

- 2n

As a consequence, 77"[z] is itself a continuous functional, as expressed in thefollowing corollary.

COROLLARY.// || z— z I < € , where z and z are arbitrary bounded functions,then

\μ*{z] ~ μ*[ z]\ <e .

It is also easy to show that the functionals Mn [ z ] and M% [ z ] which arise in

the leveling process are continuous in a similar sense.

5. The n-dimensional case. There was nothing in our treatment which de-

manded that z be a function of two variables only, or even that the variables be

numbers. Most generally we can say:

Let S i , S 2 > , Sβ be arbitrary point sets, and let z(s) be a bounded function

defined over the Cartesian product S — Sx X S 2 X * * * X S^. Let 7\, T2, V

Γj be direct subproducts of the Sj such that T{ Sp Tj unless i = /. Let Elf E2,

• , Eι be the projections of S on 7\ , T2, , T^ respectively. We now define

μ[z] = inf/ 1 , . . . f / ι sup ί € S |2 (s) ~ fi(Exs) - ••• ~

where ^ ranges over all functions defined on 71;.Our permissible lines are now replaced by a rather complicated permissible

array of points t{ ι ... j defined as follows :

(a) ί i is an arbitrary point of S

(b) to every point *ί l f...,, m there exist Z points ^ , . . . , ^ 1 ^ί1, %ίm2 » # * '

^ * i i — , i m , j , j = * i i , i m ί t * i J f * β f i = = * i i » 7 5

(d) the number of points in the set is finite

(e) if til9 9im

= ί / 1 , ,/n

t ' i e n m =n(mod2).

(This last condition is not really necessary but it serves to avoid confusion.)

In order to visualize these sets it might be well to consider the case where S

is three-dimensional Euclidean space; that is, Si is the Λ-axis, S2 the y-axis, and

S3 the z-axis. If we take 7i , T2 , Γ3 as the three coordinate planes then the per-

missible point sets consist of the vertices of closed polyhedral surfaces whose

edges are parallel to the coordinate axes. If we take 7\, T2, T3 as the three

coordinate axes then the permissible point sets consist of the vertices of closed

polyhedral surfaces whose edges are parallel to the coordinate planes.

2 0 4 S. P. DILIBERTO AND E. G. STRAUS

To each permissible point set p we now associate the functional

where the summation is extended over all the N different points of the permissible

set. If we generalize the concept of equivalence so that z ~ z whenever

z(s)-z(s) - / , ( £ i s ) + •'• +fι(Eιs) ,

then 77 [ z ] is seen to be invariant under equivalence.

The leveling process consists in the construction of the sequences

according to the following rules:

)

(j =1," , l;n = 0, l , ) .

We can agaiij define the nonincreasing sequence of nonnegative functionals

- S U PseS

and

M[z] = lim «π[z] .n-»oo

All the above lemmas and theorems remain valid under these new definitions;

and the proofs, while more difficult to state, contain essentially no new ideas.

Probably the greatest deviation from the above proofs takes place in the con-

struction of the permissible set through the leveling process in the proof of Theo-

rem 3. We shall therefore describe that process in greater detail.

Choose N so large that Λί#/+1 < M + €, and let Mtf/ + Λ /+i = M + 8;

then there exist two points tγ and ί4 in S such that Eγ £χ = E γ tϊt ι , and

a n d t h a t t h e r e e x i s t p o i n t s t ί t 2 a n d ^ 1 , 1 , 2 i * 1 $ s u c h t h a t E 2 t χ = E 2 1 1 > 2 E2 t ι $ ι

T h e n e x t s t e p in t h e l e v e l i n g p r o c e s s a d d s t h e p o i n t s £ 1 , 3 , ^1,1,3* ^1,2,39

* i , 1 , 2 , 3 9 a n ^ s o on* Af ter nl + 1 s t e p s w e h a v e t h e s e t tχ,iΪ9 9 im (m — 0 , 1 , ,

nl ij € { l , 2 , , I ] ) , w h e r e

(1 ) zMΊ(t, . ... . ) > M — (

In order to form a permissible point set we have to adjoin additional points so that

condition (b) will be satisfied. Condition (b) is already satisfied for all points

tifil9 fim withm < (n — 1)1. The number of points with (n — l ) Z < m < nl is

/ !, and is therefore independent of n. It is easy to see that by adding a fixed

finite number of points (this number A depends on k and I but not on n) we can ob-

tain a point set which satisfies condition (b). Thus the augmented point set satis-

fies conditions (a), (b), (d). Since no points of the form i,/» , , / m , ; , / O Γ ^i,/i, ,

/>ιm»/were constructed in the leveling process we can satisfy condition (c) by de-

finition. For the nonaugmented point set, condition (c) is an immediate conse-

quence of (1) if € is sufficiently small. The augmented part can be constructed so

that (c) is satisfied. We denote the nonaugmented set by p' , the augmenting set by

p".Thus we have constructed a permissible point set; if we form the corresponding

functional 7Tp [ z ] , then we have

KM I =^

3Σ zwl(*Mi, . iJ

> - (B - A)[M - (2nl+2

B

24 B -

where B — B(n) is the number of points in the permissible set, so that B(n)

with n. For a suitable choice of β and n we haveoo

Hence Theorem 3 is true in this generalized case.

6. Further discussion of the leveling process. While the leveling process

gave rise to a sequence of functions zn ~ z with l i m ^ ^ | | z n | | = μ[<z]> we were

unable to show the convergence of the functions zn. In fact, we have not yet

proved the existence of a function 2 £ C# with z ~ z and | |z | | = μ [ z ] , nor

did we investigate the rate of convergence of | |£n| | . It is the purpose of this

section to treat the last two questions.

In order to prove the existence of z we prove the equicontinuity of the se-

quence \zn\ and thus insure the existence of a convergent subsequence with a

continuous limit z . To this end we first prove :

LEMMA 5. If f%(x), f2M C C Λ and \\fx - / 2 | | < € , then

max fι(x) ~max /2OO : and

Proof. Since

min f(x) = — max Γ "~

it suffices to prove the first statement. Let

then

and hence

min f \{x) ~~ min /2OO < € .

o<*£l

) > fχ(xι) — € = max f \{x) ~~ β

max f2(x) ^ max f \{x) "" β ,

or

max fι(x) ~~ max / 2 M0<x<\ 0 l

Similarly,

max /2(*) ~ max f \{x) <

We define

n

"* ( ) = V ( )

n

hn(y) = Σ hk(y) ,

so that

= z ~ In "" Ίn

The equicontinuity of {z^l will be the direct consequence of the following

result.

THEOREM 4. If for fixed y and Δy we have

\z(x,y) -z(x,y + Δ y ) | < e /or 0 < ^ < 1 ,

then we have

\hn(y)-hn(y+Ay)\ < e (n = l ,2," ) .

Similarly, if for fixed x and ΔΛ; i^e have

\z(x,y) -z{x+άx,y)\ < e for 0 < y < l ,

then we have

< 6 ( n = l , 2 , " ) .

Proof. It obviously suffices to prove the first part of the theorem.

Let z* = z(x,y) — gn{x); then

\z*(x,y) ~ z*(x,y + Δ y ) | = U ( * f . y ) - z(x f y + Δy) | < e for 0 < x <

Hence, if in Lemma 5 we let fι(x) — z*(x9y) and f2(x) — z*(x,y + Δy), we

obtain

max [z(x,y) — gn(x)] ~ max [z(x, y + Δy) - gn(x)]0i o ς i

min [z(xty) - gπ(*)] - min [z(xty + Δy) - gn(x)]O l 0<x<,\

< 6

< e

z ~~ gn ~~ hn is vertically level. Hence

in [z{x, y)-gn(

in [z (x, yI

If

2 l

~hn{y + Δ y ) |

max [z(* f y) - gn(x)] ~" max [Z(Λ, y + Δy) - g n(*)]

mm , y) - gn(a:)] ~ min [z(x, y + Δy) - gn(x)] I < 6

The discussion so far has failed to settle the questions of the rate of con-

vergence of | | z π | | and of the convergence of zn. We were able to obtain only

partial answers. At the suggestion of the referee we omit the proofs of most of the

following statements; their sequence will have to indicate our derivation.

LEMMA 6. For n>2we have

i i g j > κ ι ι > ι i i ι ι i i ι

LEMMA 7. We have also

APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUNK OF FUNCTIONS 2 0 9

> H l w - J I +(22"+1-2)llgJv+nll

Σ ( 2 2 N ' 2 k + 1 - Dl l g^Hfcl l - Σ ( 2 2 " ~ 2 * + 2 - 1 ) I I

THEOREM 5. The norm \\z\\ satisfies

THEOREM 6. For every € > 0 there is an n0 such that for all n > n0 we have

llgn||<(2+e)W/log2n.COROLLARY. The following relations hold:

l im \\gn\\ = K " II*nII = 0 .

DEFINITION. A function z(x,y) is level if it is both horizontally and verti-

cally level.

THEOREM 7. For every z £ CR there is a Z £ CR such that Z is level9

Z ~ z, and μ[z] = \\z\\ .

Proof. According to Theorem 4, the sequence \zn} has a uniformly convergent

subsequence {zni}.het

I = lim zni .

i-oo

According to Theorem 6 we have

lim || zni + 1 — z π j l = 0

hence

z = lim z n i + i .

Of the two functions zn., -z^ i + 1 one is horizontally level, while the other is

vertically level; hence the common uniform limit is level.

Since zni ~ z, we have Z ~ z and

|| z | | = lim | | z π i | | = lim || zn\\ = μ[z] .i-*co

2 1 0 S P. DILIBERTO AND E. G. STRAUS

We remark that the bound obtained for \\gn || in Theorem 6 .does not seem to be

the best possible. In fact in all the cases we have investigated we obtained

| |gΛ | | < c 2 " " . Such an estimate would of course settle the unsolved question of

the convergence of the sequence {z Λ | .

Another unsettled question is that of the existence of a minimizing function z

equivalent to a discontinuous function z . While Theorems 4-7 remain valid with

minor modifications for discontinuous z, Theorem 4 no longer implies the exist-

ence of a convergent subsequence of {zn}.

U N I V E R S I T Y O F C A L I F O R N I A , B E R K E L E Y

U N I V E R S I T Y OF C A L I F O R N I A , L O S A N G E L E S

CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS

I. I. HIRSCHMAN, JR. AND D. V. WIDDER

1. Introduction* In the present paper we shall consider the inversion of a class

of convolution transforms with kernel G(t) of the form

(1.1)

(1.2)

(- co < t < oo) ,

*(•)= Π " " )es/bk

ak = bk + ic^ (A = 1,2, * * •) being a sequence of complex numbers such that

(1.3) Σk=ι

Σ (ck/bk)2 <oo.

This class of kernels is more extensive than that treated previously by the authors,

see [4] , [ 5 ] , [6] , and [7] however the results obtained here are slightly less

precise than those which it was possible to obtain there. We shall show essentially

that if

(1.4) /(*)= / lG(*-t)c iα(t ) ,

and if xx and x2 are points of continuity of CC(t), then

(1.5) lim /fli-oo * 1

*2 DΠ 1 - -

Here D is the operation of differentiation, and e that of translation through the

Received November 20, 1950.Pacific J. Math. 1 (1951), 211-225.

211

2 1 2 I. I. HIRSCHMAN, JR. AND D. V. WIDDER

distance l/α, so that, for example,

α2 / \ 6χ b2

bt b2

1I rll

aχa2

If we replace equation (1.2) and inequalities (1.3) by the more special relations

" / sΛ(1-6) E(s)= Π 1 - - 7 ,

00

(1.7) l im6fc/fc=Ω>0, Σ ( c * / ί > * ) 2 < 0 0 »

we have in addition the complex inversion formula,

(1.8) lim fχ*2 dx fc f(λw + x)K(w) dw = α(* 2 ) ~ Ot(%i) ,

where

(1.9) K(w) = f™ E(s)e~sw ds

and Cχis a closed rectifiable curve encircling the segment [—ίΩ , iΩ] and lying

in the strip | &w \ < Ω/λ . The inner integral in formula (1.8) is to be taken in

the counterclockwise direction.

As one example we may take

v! x Γ(l/2 + v/2)

2

Γ(l/2 + v/2 - s/2) Γ(l/2 + v/2 + s/2) '

G ( 0 = Γ(l/2+,/2)

2 '

CONVOLUTION TRANSFORMS WΓΓH COMPLEX KERNELS

where Hv > — 1 . If

213

/(«)= Ci*1*then

limR -α>

1 - = α ( χ 2 ) - α-1/2+V/2 +fe

and if Hv > 0, then

lim — — / dx I . f{x + ιkw) |_cos w] dw

See [7] and [β] , and [9] . A second example is

E(s) =π2s ΊTVcos I

2 2 2 2 2 2 2

G(t) = - cos e%(e*)77 2

for - 1 < Rz < 1. If

77

then

lim J» 1

-1/2

D

- v/2 + k1 -

~l/2 + v/2 +kj

See [ 2 ] .

2. Inversion of a class of convolution transforms. We assume as given through-

out this section a sequence, {α }J°, of complex numbers α^ = b^ + j'c^ subject

to the restrictions

(2.1) Σ (l/bk)2

k=i

We define the entire functions

(2.2) EΛ,n(s) =

Σ (tk/bk)2 <<».

k=i

= Π (I ~ s/ak)k=m+l

= Π

The definition of Em(s) is significant because

00 / s \

π i--••

00

exp ΣΛ + l -f- i

and because the series Σ ^ + 1 | a^ \ ~2 , Σ ^ + 1 c^/bj^b^ -f ί'c^) converge as a

consequence of (2.1) and Schwarz's inequality. Similarly, Fm(s) is well defined.

We define

(2.3)

We also set

(2.4) βχ (m)

THEOREM 2a.

1

(*=0, l , ) .

β2(m) = m in (bk,6o

, m yjf ± , z,, ) ,

CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 215

then we have

A.

β.

C

D.

f_m \Gm(t)\e-σt dt <

P«(fl)G0(t)=G«(t);

(d/dt)kGm(t)=O(e^t), *-»+«

= 0,l, ) ,

Conclusion A is an immediate consequence of Hamburger's theorem; see

[4, pp. 141-144]. We define g(u) = e*" 1 for -co 0, and for b^ < fts < oo if b^ < 0. Let

g i * g 2 ( O = Xoo gi (ί "" u)g2 (u) si" ,

and so on; then by the convolution theorem for the bilateral Laplace transform

we have

£ ^ g»+i * g«+2 * •'• *gn(t)e-stdt = [ £ . , „ ( « ) ] " *

for βι(m) < Rs < β2(m). From the complex inversion formula for the bilateral

Laplace transform we obtain

*g«+2 * ••• * g » ( t ) =,2πi

Since

= 0

for —oo < t < oo , it follows that

* ••• *gn(t) =Gn(t) (-oo< ί <oo).n-oo

See [4; pp. 139-145]. It is easily seen that

SZ kk(t)\e~stdt = [(1 - 8/bkyfo\bk/ak\Tι ,

for -oo < Hs < bk if bk > 0, or for bk < Hs < oo if bk < 0. By Fatou's

lemma we have

f_l K(t)\e-σt dt < lim inf j Γ | g l l + ι * * gn(t) \e~σt dt ,

< lim inf X " |g B + 1 | * ••• * | g π ( ί ) | e " σ t d t

so tnat conclusion B is established.

Conclusion C follows from the identity

P»(D)est = e s t f [ ( 1 -

Conclusion D may be established by shifting the line of integration in the integral

defining Gm(t) to Rs = yx and Hs = γ2 . See [4; pp. 152-154] .

In what follows we shall write G(t) for G0{t).

THEOREM 2b. //

(a) G(t) is defined as in Theorem 2a,

(b) βx(o) < c < /32(o), c + n > A(o), c + γ2 < 02(o),

(c) α(ί) is of bounded variation on every finite interval, d(t) — 0(e7i ) as

t —•> - <» , α(ί) = 0(e7^) as t —> + oo $

(d) flnΦ) is defined as in equation (2.3),

(e) / ω = / ω G ( * - ί ) β

c l d α ω ,

(f) ^i and %2 are points of continuity of 0C(ί),

then

lim = a(xa) - α (

From assumption (c) and from conclusion D of Theorem 2a we may show, using

integration by parts, that each of the integrals

converges uniformly for x in any finite interval. Since Pm(D)G(t) = Gm(t) by con-

clusion C of Theorem 2a, it follows (see [4; pp. 167-170]) that

(2-5) P. &>)/(*)= f_lGΛ(χ-t)ectda(t)

Multiplying by e~cx and integrating by parts, we have

(— 00 < x < OO) ,

α(t)dt

a(t)dt.

Since this integral converges uniformly for Λ; in any finite interval, we obtain

e-c*PΛφ)f(x)dx

Γ2 dx foo I 3

oo I ->„ *>] α ( t ) Λ

218 1. I. HIRSCHMAN, JR. AND D. V. WIDDER

»i α(t )Λ

We thus need only show that if x is a point of continuity of α(ί) we have

(2.6) lim fW Gm(x - t)β-e<*-*> α ( t ) dt = α(x) .

We shall first show that for any 6 > 0 we have

(2.7) lim G«(t)e- C t α (x - t) dt = 0 .

Using assumptions (a) and (b) we see that it is enough to prove that for any

δ with βiiO) < δ < /32(0), we have

0 so small that ^ ( 0 ) < 8 - 2η < 8re

(2.8)

Choose rj >

\t\ > 6 we have

"" (sinh

so that it is enough to prove that

lim /°° \Gn(t)\e~$t [sinh ηt]2dt = 0

)S2(0). For

Using conclusions A and B of Theorem 2a we see that

C \G.W\e-* l,iihηtl'dt

F . ( 8 - 2 τ > ) Em(B)

and equation (2.8) follows from this. We assert that

(2.9) lim £ Gm(t)e'ct dt = 1

(2.10) lim^sup f_l \Ga(t)\e-ctdt =

These results are immediate consequences of conclusions A and B of Theorem 2a.

Now x being fixed and 77 > 0 being given, let us choose e > 0 so small that

I (X(t) — a(x) I <_ 77 for I ί — x I < € . We have

j Γ G Λ s - O ^ ^ α ί O d t - <*(*) =ίi +12 +I3 ,

where

h =

[a(x~-t)-a(χ)]dt

[a.(x-t)-a(x)]dt

We have limm-oo/i = 0 by equation (2.9), limm-»co/2 ~ 0 by equation (2.7), and

lim supm-»oo|/3 j <. 77 by equation (2.10). Since 771s arbitrary our demonstration is

complete.

3. Complex inversion formulas* In this section we restrict our attention to a

much more special class of kernels. We suppose that

(3.1)

We define

(3.2)

(3.3)

bk>0, bk~Ωfc

77 IB—

H(λ,s) =

E(s) = Π

λ2 + (1-λ 2 ) \ak\bk

bl - s2 (0 < λ < 1)

The product (3.3) is defined for s f^ h^ {k — 1,2, •) since it can be rewritten

ΠH{λ,s) =

i-^-+(i-λ2)\<*k\ ~ bk

π

and assumption (a) implies that ΣJ° [ | α& | — i^] ^^""ι i s convergent. We define

(3.4) β = m i n 6fe .

THEOREM 3a. //

1 ri«> β^£(λs)

2τri E(s)ds

then

A. G(λ,w) is analytic for \&w\ < Ω(l - λ)

(d/dw)kG(\,w) = 0(e 7iu) (u

(0 < λ< 1) ,

= 0 , 1 , . . . ) ,

where yι > —β, γ2 < β, uniformly for \v\ < Ω ( l — λ ) "~" 6 , e > 0.{Here

w — u + iv.)

• ' - o o<//(λ,σ), - β<σ< β.

We shall write G(ί) for G(0, ί) .

We assert that

(3.5) log \E(σ + i r ) I ~ Ω | τ I (r —> ± oo)

uniformly for σ in any finite interval. We define

00

We have

from which it follows that

E(s) » 4(3.6) lim — = Π Hr

uniformly for 0 < 6 < | arg s\ <π — € . From [ l , pp. 267-279] we have that

log |£* (cr + ir) ~ Ω | τ | as r —> ±°°, uniformly for α in any finite interval.

Relation (3*5) now follows.

Conclusion A follows immediately from (3.5) and the definition of G( h,w).

Conclusion B is a consequence of (3.5) and Hamburger's Theorem. The two con-

clusions C are obtained by shifting the line of integration in the integral defining

G(λ,ί) to Rs = 7i , and to Hs = γ2 , respectively. See [6, pp.688-691]. To

establish conclusion D we introduce the functions

Π i-^Hγ.

Π - ik=i \ akl

It is immediate that

lim G π (λ,ί) = G(λ,t) ( - oo < t < oo) .

222

We define

I. I. HIRSCHMAN, JR. AND D. V. WΓDDER

where j(t) = 0 for -co < t < 0; ;(0) = 1/2; j(t) = 1 for 0 < t < oo.

It is easily verified that for ~~&£ < cr < b^ we have

- λV/α?

Just as in §2 we may show that

G n ( λ , ί ) = lim — [/ii(λ,t) * •••* hn(λ,t) * h π + i ( 0 f t ) * •••* h«(θ f t ) ] .w- oo at

Here h1*h2(t) ~ X°^ Aι(ί ^u)dhι(u). Note that this differs from the convention

employed in §2. By Fatou's lemma,

< lim inf f™ e'σt \dhι(\,t) * *hn(λ,t) *hn+1(O,t) * •• *Λm(θ,t)

< lim inf Π 'C

<Π Π

By Fatou's lemma, again,

<π

< Π λ2 + ( 1 - λ 2 )\ak\bk

CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 2 2 3

This completes the proof of the theorem.

We define

(3.7) K(w) = f* E{s)e'sw ds .

It follows from relations (3.1) that given € > 0, for all sufficiently large r wehave log | £ * (reiθ) | < (e + | sin θ\ )Ωr. See [l, pp.267-279] . Fromequation

(3.6) it follows that

log \E{reiθ)\ < (e + |sin 0 | ) Ω r

for r sufficiently large. Using this inequality and rotating the line of integration

in the integral defining K{w) we can show that K(w) is analytic and single valued

in the w -plane except on the segment [— iΩ , iΩ] . It may also be shown, see

[l, pp.295-311] , that if C is a closed rectifiable curve encircling [~iΩ , iΩ]

then

(3.8)

the integration proceeding in the counterclockwise direction.

LEMMA 3b. If C\ is a closed rectifiable curve encircling [—ίΩ , iΩ]and

contained in the strip \v\ < Ω/λ , then

fc^G(\w+x-t)K(w)dw=G(\,x-t),

the integration proceeding in the counterclockwise direction.

We have

1

2 771- / G(\w+χ-t)K(w)du

E(λs)

2771 E(s)ds

THEOREM 3 C //

(a) G(t) is defined as in Theorem 3a

(b) -β < c < β, -β < c + yι , c + γ2 < β

(c) α(ί) is of bounded variation on every finite interval and

α ( t ) = (βn*) ( t_»+oo), α ( t ) = (ey2<) (t_»-oo)

(d) f(w) = Jl0^ G(«; - ί)e c ί r fα(ί)

(e) Jζ(ιt ) is defined as in equation (3.7)

(f) C\ is defined as in Lemma 3b

(g) %ι and x2 are points of continuity of CX(ί), then

lim f*2 e~cx dx - i - Γ /(λ. + ^ W ώ ^ α f c l - α f e ) .λ - l - x l 277 I ^ λ

It follows from assumption (c) and from conclusion C of Theorem 3a that the

integral defining f(w) converges uniformly for w in any compact set contained in

the strip I AM; I < Ω . Hence

x)K(w)dw2πi

by Lemma 3b. The proof may now be completed exactly in the manner of Theo-

rem 2b.

4. Remark. If it is assumed that the roots of E(s) occur in conjugate pairs,

then equation (1.5) can be established under conditions less restrictive than (1.3).

A discussion of this case is given in the Master's thesis of Mr. A. 0 . Garder [3],

written under the direction of one of us.

CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 2 2 5

R E F E R E N C E S

1. V. Bernstein, Lemons sur ίes progres recents de la theorie des series de Dirichlet,Paris, 1933.

2. R. P. Boas, Jr., Inversion of a generalized Laplace integral, Proc. Nat. Acad. Sci.U. S. A. 28 (1942), 21-24.

3. A. O. Garder, The inversion of a special class of convolution transforms, Master'sThesis, Washington University, 1950.

4. I. I. Hirschman, Jr. and D. V. Widder, The inversion of a general class of convo-lution transforms, Trans. Amer. Math. Soc. 66 (1949), 135-201.

5. , A representation theory for a general class of convolution transforms,Trans. Amer. Math. Soc. 67 (1949), 69-97.

6. , Generalized inversion formulas for convolution transforms, Duke Math.J. 15 (1948), 659-696.

7. , Generalized inversion formulas for convolution transforms, II, DukeMath. J. 17 (1950), 391-402.

8. H. Pollard, Studies on the Stieltjes .transform, Dissertation, Harvard; Abstract48-3-117, Bull. Amer. Math. Soc. 48 (1942), 214.

9. D. B. Sumner, An inversion formula for the generalized Stieltjes transform, Bull.Amer. Math. Soc. 55 (1949), 174-183.

10. D. V. Widder, The Laplace transform, Princeton University Press, Princeton, 1941.

WASHINGTON UNIVERSITY

HARVARD UNIVERSITY

A THEOREM ON RINGS OF OPERATORS

IRVING K A P L A N S K Y

1. Introduction. The main result (Theorem 1) proved in this paper arose in

connection with investigations on the structure of rings of operators. Because of

its possible independent interest, it is being published separately.

The proof of Theorem 1 is closely modeled on the discussion in Chapter I

of [3] . The connection can be briefly explained as follows. Let N be a factor of

type l i t ; then in addition to the usual topologies on /V, we have the metric defined

by [ [^ l ] ] 2 = T(A*A), T being the trace on N. Now it is a fact that in any

bounded subset of N, the [[ ]]-metric coincides with the strong topology—this is

the substance of Lemma 13.2 of [3] . In the light of this observation, it can be

seen that Theorem 1 is essentially a generalization (to arbitrary rings of operators)

of the ideas in Chapter I of [3] .

Before stating Theorem 1, we collect some definitions for the reader's con-

venience. Let R be the algebra of all bounded operators on a Hubert space // (of

any dimension). In R we have a natural norm and *-operation. A typical neighbor-

hood of 0 for the strong topology in R is given by specifying e > 0, ξ 1, ,

ζn C H, and taking the set of all A in R with | |>4^ j | | < e; for the weak topol-

ogy we specify further vectors Tjί9 , T)n £ // and take the set of all A with

I {A ξi, Ύ)ι)\ < e. By a *-algebra of operators we mean a self-adjoint subalgebra

of/?, that is, one containing A whenever it contains A; unless explicitly stated,

it is not assumed to be closed in any particular topology. For convex subsets of

R, and in particular for subalgebras, strong and weak closure coincide [2,Th 5]

An operator A is self-adjoint if A* — A, normal if A A —A A, unitary if A A*

— A A — the identity operator /.

2. The main result. We shall establish the following result.

THEOREM 1. Let M, N be *-algebras of operators on Hubert space, M C N9

and suppose M is strongly dense in /V. Then the unit sphere of M is strongly dense

in the unit sphere of N.

Received October 6,1950. This paper was written in connection with a research projecton spectral theory, sponsored by the Office of Naval Research.

Pacific J. Math. 1 (1951), 227-232.

227

228 IRVING KAPLANSKY

We shall break up the early part of the proof into a sequence of lemmas.

Lemma 1 is well known and is included only for completeness.

LEMMA 1. In the unit sphere of R> multiplication is strongly continuous, joint-

ly in its variables; and any polynomial in n variables is strongly continuous,

jointly in its arguments.

Proof* It is easy to see that multiplication is strongly continuous separately

in its variables, even in all of R. Consequently [ l , p.49] we need only check

the continuity of AB at A — B — 0. Since \\A\\ < 1, this is a consequence of

\ \ A B ξ \ \ < \\A\\ \ \ B ξ \ \ < \ \ B ξ \ \ .

Since addition and scalar multiplication are continuous (in all of R), the con-

tinuity of polynomials follows.

The precaution taken in the next lemma, in defining the mapping on the pair

{A,A ), is necessary since A —> A is not strongly continuous.

LEMMA 2. Let f(z) be a continuous complex-valued function, defined for

\z\ < 1. Then the mapping (A, A*) —> f(A) is strongly continuous on the

normal operators of the unit sphere of R,

Proof. We are given a normal operator Ao with 11 >4oil S 1> a positive e ,

and vectors ξ( in // (i = 1, , n). We have to show that by taking A, A* to be

normal with norm < 1, and in suitable strong neighborhoods of AQ, AQ, we can

achieve

ω l | [ / u ) - / U o > ] £ . - l l < e .

By the Weierstrass approximation theorem, there exists a polynomial g in two

variables such that

(2) |g(z,z*) " f(z)\ < e / 3 ,

for \z\ < 1, z* denoting the conjugate complex of z. By elementary properties of

the functional calculus for normal operators, we deduce from (2):

(3) \\giAtA*)-fU)\\ < e / 3 ,

(4) \\g(Λo,A*)-f<Ao)\\ < e/3.

By Lemma 1, if we take A, A* in appropriate neighborhoods of AQ, A%, we have

A THEOREM ON RINGS OF OPERATORS 2 2 9

(5) \\[gU,Λ*)- g(AΰtAt)]ξi\\ < e/3.

By combining (3), (4), and (5) we obtain (1).

The next lemma follows from Lemma 2 as soon as it is admitted that * is

strongly continuous on unitary operators. This can, for example, be deduced from

two known facts: (a) the strong and weak topologies coincide on the set of unitary

operators, and (b) * is weakly continuous.

LEMMA 3. Let f be a continuous complex-valued function defined on the

circumference of the unit circle. Then the mapping U —> f(U) is strongly con-

tinuous on the set of unitary operators.

The Cayley transform is the mapping A —> (A — i)(A + i) 1 ; it is defined

for any self-adjoint operator and sends it into a unitary operator.

LEMMA 4. The Cayley transform is strongly continuous on the set of all self-

adjoint operators.

Proof. We have the identity

(6) 04 - i)(A + iΓι - U o - i)(A0 + iΓι = 2i(A + i)'1 (A - AQ)(A0 + i)~ι .

When A is self-adjoint, we have | | (A + i)~ ι | | < 1. In order to make the left side

of (6) small on a vector ξ, it therefore suffices to make A — Ao small on the

vector 04 0 + i) ι ξ.

We shall prove a stronger form of Lemma 5 below (Corollary to Theorem 2).

LEMMA 5. Let h be a real-valued function defined on the real line, and sup-

pose that h is continuous and vanishes at infinity. Then the mapping A —* h(A)

is strongly continuous on the set of all self-adjoint operators.

Proof. Define

f{z) = h[-i(z + 1)U ~ 1 Γ 1 ] for \z\ = 1 , z φ\%

=^0 for z = 1.

Then f is continuous on the circumference of the unit circle. Moreover,

h(A) = f[(A - i)U + i ) " 1 ] .

The mapping A —* h(A) is thus the composite of two maps: the Cayley transform,

2 3 0 IRVING KAPLANSKY

and the mapping on unitary operators given by /. By Lemmas 4 and 3, these latter

two maps are strongly continuous. Hence so is A —» h(A).

Proof of Theorem 1. There is clearly no loss of generality in assuming M and

N to be uniformly closed, for the unit sphere of M is even uniformly dense in the

unit sphere of its uniform closure.

Let us write Z for the set of self-adjoint elements in M, and Zγ for the unit

sphere of Z. Let B be a given self-adjoint element in N, | | B | | < 1. By hypothe-

sis, B is in the strong closure of M. We shall argue in two successive steps that

B is actually in the strong closure of Z\ We begin by remarking that B is in the

weak closure of M, since the latter coincides with the strong closure of M Now *

is weakly continuous, and hence so is the mapping A —> (A + A*)/2 This

mapping leaves B fixed, and sends M onto Z; hence B is in the weak closure of

Z. Since Z is convex, this coincides with the strong closure of Z.

Let h(t) be any real-valued function of the real, variable t which is continuous

and vanishes at infinity, satisfies | h (t) \ < 1 for all t, and satisfies h(t) —t

for I ί I < 1. We have that h(B) — B. Also h can be meaningfully applied within Z,

since we have assumed M to be uniformly closed, and in fact h(Z) — Zχ> By

Lemma 5, the mapping A —* h (A) is strongly continuous on self-adjoint oper-

ators. Hence B is in the strong closure of Z ι .

This accomplishes our objective as far as self-adjoint operators are concerned.

To make the transition to an arbitrary operator, we adopt the device of passing to

a matrix algebra.1 Let N2 be the algebra of two-by-two matrices over N. In a

natural way, TV 2 is again a uniformly closed *-algebra of operators on a suitable

Hubert space (compare §2.4 of [3] )• It contains in a natural way M2, the two-by-

two matrix algebra over M. The strong topology on N2 is simply the Cartesian

product of the strong topology for the four replicas of N; thus M2 is again strongly

dense in N2 Now let C be any operator in N, \\C\\ < 1. We form

D -

and we note that D £ /V2> 0 * ~ D9 WOW < 1. Let U be any proposed strong

neighborhood of D. By what we have proved above, there ex i s t s in U a self-adjoint

element F ,

1 1 am indebted to P. R. Halmos for this device, which considerably shortened rayoriginal proof of Theorem 1.

A THEOREM ON RINGS OF OPERATORS 231

_ /G ΛF" V *)

with F G M29 | | F II < 1. By suitable choice of U we can make H lie in a given

strong neighborhood of C. Also | | F | | < 1 implies \\H\\ < 1. This proves that C

lies in the strong closure of the unit sphere of M, and concludes the proof of

Theorem 1.

3 Remarks, (a) Since strong and weak closure coincide for convex sets, we

can, in the statement of Theorem 1, replace "strongly" by "weakly" at will.

(b) From Theorem 1 we can deduce that portion of [2, Th.8] that asserts

that a *-algebra of operators is strongly closed if its unit sphere is strongly

closed; but it does not appear to be possible to reverse the reasoning.

(c) As Dixmier has remarked [2, p. 399], Theorem 1 fails if M is merely

assumed to be a subspace (instead of a *-subalgebra).

4. Another result. In concluding the paper we shall return to Lemma 5 and

show that the hypothesis can be weakened to the assumption that h is bounded

and continuous. It should be noted that we cannot drop the word "bounded," since

for example it is known that the mapping A —> A is not strongly continuous.

Actually we shall prove a still more general result, which may be regarded

as a generalization of Lemma 4.2.1 of [3]

THEOREM 2. Let hit) be a bounded real-valued Baire function of the real

variable tf and Ao a self-adjoint operator. Let S be the spectrum of AOf and T the

closure of the set of points at which h is discontinuous; suppose S and T are

disjoint. Then the mapping on self-adjoint operators, defined by A —> h(A)9 is

continuous at A = Ao.

Proof. We may suppose that

(7) |A(ί) | < 1

for all t. Given 6 > 0, and vectors ξ^% we have to show that for A in a suitable

strong neighborhood of Ao , we have

(8) \\ίh(A)-h(A0)]ξi\\ <e.

Choose a function k (t) which satisfies: (a) k is continuous and vanishes at in-

finity, (b) k(t) = 1 for t in S9 (c) k(t) = 0 for t in an open set containing Γ. Define

232 IRVING KAPLANSKY

p — hkf q — 1 ~~ k ~f" hk* Then p = q = A on S, and so

(9)

Also p and ςr — 1 are continuous and vanish at infinity. Hence Lemma 5 is appli-

cable, and for a certain strong neighborhood of Ao we have

(10) II [p(A)-p(A0)]ξi || <e/4, || [q(A) - q(A0)] ξt \\ < e/2 .

The following is an identity:

(11) h = (l-h)p + hq.

From (9) and (11) we get

(12) h{A) ~h(A0)= [1 - A M ) ] [ P U ) - p U o ) ]

From (7), (10), and (12), we deduce (8), as desired.

If in particular h is continuous, then Γis void and we get a simplified corollary.

COROLLARY. Let h(t) be a continuous bounded real-valued function of the

real variable t. Then the mapping A —> h(A) is strongly continuous on the set

of all self-adjoint operators.

REFERENCES

1. N. Bourbaki, Elements de Mathe'matique, Livre III, Topologie Generate, ActualitesSci. Ind., No.916, Paris, 1942.

2. J. Drxmier, Les fonctionelles lineaires sur I'ensemble des operateurs homes d'unespace de Hubert, Ann. of Math. 51 (1950), 387-408.

3. F . J. Murray and J. von Neumann, On rings of operators /K, Ann. of Math. 44 (1943),716-808.

UNIVERSITY OF CHICAGO

AN ITERATIVE METHOD FOR FINDING CHARACTERISTIC

VECTORS OF A SYMMETRIC MATRIX

W. KARUSH

1. Introduction* Given a real symmetric linear operator A on a vector space 8,

we wish to describe a procedure for finding a "minimum" characteristic vector

of A, that is, a characteristic vector with least characteristic value, supposing

such to exist. The method to be used is, in a general way, the following. Select

an initial vector x° and a positive integer s > 1. Imbed x° in an s-dimensional

linear subspace 8° (appropriately selected). Determine the next approximation xι

as the minimum characteristic vector relative to this subspace (to be defined

later). Next, imbed Λ;1 in an s-dimensional subspace 8 and determine x2 as the

minimum characteristic vector relative to this subspace. Proceeding in this manner,

construct a sequence of subspaces S°, S 1 , of fixed dimension s, with a

corresponding sequence of vectors xι,x2, . It is to be expected that under

appropriate hypotheses the sequence of vectors will converge to a minimum char-

acteristic vector of A.

We shall treat the case when 8 is of finite dimension n, and 8 ι is chosen as

the subspace spanned by the vectors χι, Ax1, A2xι, , As~ιxι We shall es-

tablish the desired convergence under these circumstances, the sequence {x1}

satisfying at the same time a relation xι + i = xι + Ύ]1 with (xι, Tjι) = 0. The main

result is formulated in Theorem 2 of §6. An analogous result holds for a "maximum "

characteristic vector.

It is of interest to compare the present iteration method with what might be

called Rayleigh-Ritz procedures. In the latter, one fills out the space 8 by a

judiciously chosen monotone sequence of subspaces

8! C 8 2 C S 3 C . . . (dim 8; = i)

of increasing dimension. One then obtains successive approximations to a mini-

mum vector of A by determining minimum characteristic vectors of the successive

Received May 20, 1950. The preparation of this paper was sponsored (in part) by theOffice of Naval Research.

Pacific J. Math. 1 (1951), 233-248.

233

2 3 4 W. KARUSH

subspaces. This procedure has the serious computational drawback that to obtain

an improved approximation a problem of increased complexity, that is, of higher

dimension, must be solved. This restriction is important even in the finite dimen-

sional case where the iteration, in theory, terminates in a finite number of steps.

The method of the present paper, however, requires only the solution of a problem

of fixed dimension s at each step, the dimension s being chosen from the outset

as any desired value. The 8* form a chain of subspaces in which successive sub-

spaces 8* and 8 ι + 1 overlap in xi+ι in general this chain will be infinite even

when 8 is finite-dimensional. Thus the method is useful where it is desired to fix

beforehand the degree of complexity for all steps; and yet a great many iterations

may readily be performed. This is the case with high speed computing machines.

The present procedure may be interpreted as a gradient method; cf [ l ] . For

s = 2, in the equation xι+ι = xι + rj1, η* is a multiple of the gradient at x = xι

of the function Gc, Ax)/(x, x) For 5 > 2, the vector r] contains higher order terms.

The applicability of the present procedure with s = 2 to quadratic functionals in

infinite-dimensional spaces has been pointed out to the author by M.R.Hestenes

in conversation, and has been outlined by L. V.Kantorovitch [2] .

2 Subspaces. Before describing in detail the iteration procedure to be used,

and proving its convergence, we find it convenient to formulate some preliminary

results. In this section we construct an orthogonal basis for the space spanned

by the powers of A operating on a fixed vector x; in the next section we describe

the characteristic roots and vectors relative to certain subspaces of this space.

We shall encounter polynomials py(λ) of central importance. In these two sections

we shall be treating, essentially, only one level of the iteration. Accordingly, the

superscript i denoting the various steps of the iteration will not appear until §4,

where we are concerned with the progression from one level to the next.

Let S denote the ^-dimensional space of π-tuples of real numbers; by vector

we understand always an element of 8. We consider a linear operator A on 8 which

is real and symmetric; that is, one for which Ax is a real vector and

(Ax, z) = (x,Az)

for arbitrary real vectors x, z. A characteristic number (root, value) of A is a

number λ for which there exists a non-null vector y such that

Ay = λy .

There are n (real) characteristic numbers (counting multiplicities).

CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 2 3 5

With a non-null vector x we associate the number

( . (x,Ax)

(pc,x)

and the vector

ξ{x) =Aχ- μ{x)x .

Let λ m j n ( λ m a x ) be the least (greatest) characteristic root, of A. It is well known

that

(1) λ m i n = min μ(x) , λ m a x = max μ(x) , (x £ £ ) .xέo xέo

For a non-null vector x we define the subspaces

< l j ( * ) = (x,Ax, ~>,A>-ιx) (j = 1,2,3, • • • ) ,

α ( x ) = (x,Ax,A2x, •••) ,

where, in each case, the right side of the equation denotes the space spanned by

the designated vectors. The space U (x) is the smallest invariant subspace con-

taining x denote its dimension by r = r(x) Clearly dι C fl2 C C flr = fl,

where " C " denotes strict inclusion. The space CL contains r independent char-

acteristic vectors of A. We now construct an orthogonal basis for Gy .

LEMMA 1. Let the vectors ξj (j — 0,1, , r) be defined by

(2) £o=x, £i=Aξo- ( ())

ξ Aξj-μjξj- tj

ti = TT^T U = 1,2, ••-, r - 1) .I b l

/or /, A = 0,1, , r ~ 1, zi e Aαve ζj φ 0,

(3) Gj + iGO = ( o , ^ i , * '* , ξj) , (ξj , ξk) = 0 ,

The lemma may be verified directly by induction. We remark that ξΓ — 0.

236 W. KARUSH

LEMMA 2. Let the polynomials py(λ) (/ = 0 , 1 , , r) be defined by

Po(λ)=l, pi(λ) = ( λ - μ 0 ) , PaOO = (λ-μo)(λ-/*i) ~ *? .

PJ + ι(λ)=P J-(λ)(λ-^.)-t/p J..1(λ) 0 = 1,2, — , r - 1 ) .

Suppose B is an invariant subspace containing x; write

(4) x = myi + a 2 y 2 + ••• + aι yι

in terms of a basis of characteristic vectors of B Then

(5) £ ; = a ^ y C λ J y ! + a 2 p ; ( λ 2 ) y 2 + ••• + aιpj(λι)yι ( j = 0,1, •••, r ) ,

where λ k is the characteristic number of yk

The lemma follows immediately from the definitions (2).

The polynomials py(λ) have also been used by C. Lanczos [3]

3 Characteristic values relative to subspaces Let B be an arbitrary (linear)

subspace of S; let 7Γ be the operator on S which carries any vector into its pro-

jection on B. We define a linear operator 4 (B) on B to B as follows:

Then A(Q) is a symmetric operator on B, since 4(B) = πAπ. By the characteristic

roots and vectors of A relative to the subspace B, we mean the corresponding

quantities of 4(6). If B is invariant, then these quantities are characteristic for A

itself. We shall use the following easily verified fact: y is a characteristic vector

relative to B with characteristic value λ if and only if y φ 0, y G B, and (Ay, z)

= λ(y, z) for z C B. By a minimum characteristic vector of B we shall mean a

characteristic vector relative to B with least characteristic value. When no con-

fusion can arise we shall omit the qualifying term "relative/*

LEMMA 3. The j characteristic roots relative to the subspace dj(x) are dis-

tinct and are given by the solutions of

p7 ( λ ) = o .

Each characteristic vector (relative to fly) has a non-null projection on x.

To prove the last statement, suppose that γ is a characteristic vector with

characteristic value λ . If (y, x) = 0, then (y, Ax) ~ (Ay, x) = \(y, x) = 0, and

CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 237

(y, A2x) = (Ay, Ax) = λ (y, Ax) = 0, , and (y, A^"ιx) = O From the definition

of Gy it follows that y is orthogonal to this space. But y belongs to this space;

hence y = 0, a contradiction.

The distinctness of the roots now follows. For if two independent characteristic

vectors belong to λ then there is a non-null linear combination orthogonal to x

belonging to λ.

To complete the proof we use the basis (3) of Uy . The matrix representation,

call it Ajy of A(&j) relative to this basis has as element in the (k + l)st row and

(Z + l)st column;

(Aξk,ξι), - i)

Using (2) and the second line of (3) we find that

Ai =

Mo

* 1

0

•

•

* 2

0

o* 2

μ-2

• •

•

0

••• tj-i

Thus, the characteristic roots are the roots of the polynomial

qj(λ)= \\Ij-Aj\ ,

where /y is the /-rowed square identity matrix. Let q$0\) = 1. Direct calculation

shows that qι (λ) = pί (λ), and that the ^y(λ) satisfy the same recursion relation

as the pj(\). Hence the two sets of polynomials are identical. This completes

the proof.

LEMMA 4. Let Vj be the minimum characteristic root relative to &y; that is,

Vj = min. root of pj (λ) (j = 1, 2, ••• , r) .

Then

(6) = vr

2 3 8 W. KARUSH

where \χ is the minimum characteristic root of the invariant subspace uΓ . Further,

each root cr of each polynomial pj(\) satisfies

( 7 ) Kin < σ < λin < σ < λ m a x

The last statement follows at once from Lemma 3 and (1) when we notice that

each characteristic root σ is a value of μ(z) = (z, Az)/(z, z); namely, σ is that

value obtained by replacing z by the corresponding characteristic vector.

To prove (6) we apply (1) to the operator A(dj). Using the fact that (Az, z)

= [A (&j)z, z ] for z in Gy, we find that

= min μ(z) , (z in CL ) .J

From U. d U.-+! we infer that the roots are non-increasing. Suppose that v^

= v\t\\ . Denote the common value by v. From the recursion formula for the poly-

nomials it follows that

contrary to the definition p 0 (λ) = 1.

LEMMA 5. The minimum characteristic vector relative to Uy is given by

where

More generally, the characteristic vector belonging to an arbitrary root σ is

obtained by replacing vy by σ on the right in (8). To prove this, let z denote the

vector obtained by this substitution. It is sufficient to show that η = Az — σz

is orthogonal to Cίy; to this end we use the basis in (3). Using the definition of

z and the relations (2) and (3), we find that

-(σ- μo)]\x\2=O,


τί-i Tl

for

obtain

= — V [PI + I M - ίPiWCσ -Mz) - P H W * ! 2 ! ] =0

Z = 1,2, , j — %. For I — j — 1, the term in p/+t does not appear, and we

ain

I 2

- pj(σ) = 0 .

This completes the argument.

4 The iteration procedure. We shall henceforth be dealing with a sequence

\x } of vectors; with each vector we associate the quantities described previ-

ously for an arbitrary vector x. To indicate dependence upon xι we shall adjoin

the superscript i to the symbols denoting these quantities.

Consider an initial vector x° φ 0. By definition r° [= r{x0)] is the dimension

of u [= d(x )] , the smallest invariant subspace containing #°. Since U =

dr0 (x°), according to Lemma 3 there are r° distinct characteristic roots

λ0

relative to Q°; and the corresponding characteristic vectors can be normalized

so that

X = Y i + Yo + •••-!- y r o

All vectors considered below will lie in the invariant space G° Henceforth the

symbols λy and yj will denote the characteristic quantities of this subspace.

To specify the iteration procedure at hand we require, besides x°,the selection

of a fixed dimension s > 1. We remark at this point that the significant case is

that for which the dimension of the invariant space &(xι) at every stage exceeds

s; that is,

2 4 0 W. KARUSH

(9) (i = 0,1,2, •••) .

To simplify presentation, unless otherwise stated it will be assumed that this

condition holds. The trivial case in which (9) fails will be treated at the end of

this section.

Consider now the s-dimensional subspace Q^ = Us (x°) Relative to this sub-

space there is, by Lemma 3, a unique minimum characteristic vector x° -f rj° with

(x°, τ}°) = 0; call it xι. Now form &\ = ds (xι) and select x2 as the unique mini-

mum characteristic vector relative to this space of the form xι + η ι , (χι

9 rjι) = 0.

In general we define # ι + ι as the minimum characteristic vector xι + Tjι, (xι, Ύ]1)

= 0, relative to the subspace CL£. Notice that these subspaces form a chain in

which successive subspaces of index i and i + 1 overlap in xι ι .

LEMMA 6. The sequence \x \ is given by

(τ,')2 4 l h i ,) 1 ί s ' '

where vι is the least root of p | (λ). Further,

(ID v* = μ ( * i + i) .

i4Zso {v } is decreasing; in fact

(12) λj < vi = v/ < z^-x < ••• < vί = μ{xl)9

where v\ is the minimum zero of ρι. (λ).

By Lemma 3 the minimum characteristic root relative to &ι

s is Vs\ It follows

by the definition of # l + l that the equality (11) holds. The relations (12) follow

from Lemma 4, condition (9), and definition. The formula (10) is (8) of Lemma 5

interpreted for x = xι and / = s.

LEMMA 7. /rc ierms of the characteristic basis of Q° we have

(13) * i = oίy, + ( 4 y 2 + + α ^ y r 0 ,

(14) ξj = a\ pfiλjyi + 4pj(λ2)y2 + ~ + a*op/(\.o)yro

( ί = 0 , 1 , 2 , •••; j = 0 r l , *•• , r l ) .

where

(15) 1 +pjj(λfc)

+ +

Furthermore, α^ = 1

(16) 1 = α j < αf < α? < ••• .

Formula (14) follows from (13) by Lemma 2; (15) is a consequence of (13), (14)

and (10) of Lemma 6. To prove (16) we notice that pι. (λ) (j = 1,2, , s — 1)

is not zero, and has the same sign, at Xi and at vι [since by (12) the least root

of the polynomial exceeds these values] . Hence each term in braces in (15) is

positive; this completes the proof.

We conclude the present section with a consideration of the possible failure

of (9). Suppose that for some first value m of i this inequality fails. Then Cί™ is

an invariant subspace, and the minimum characteristic vector xm ι relative to

this subspace is a characteristic vector of A. Thus Gj?1"1"1 is a one-dimensional

invariant subspace containing only multiples of xm ι . It follows that xι = xm ι

for i > m + l But the argument used in establishing (16) shows that xι — Lyχ9ft

L > 0, for i > m + 1. The theorems to be proved in the next two sections now hold

trivially. We are thereby justified in the assumption of (9).

5 Convergence in direction. We shall first prove that the sequence \xι\

converges in direction; in §6 we shall establish the more troublesome property

of convergence in length.

THEOREM 1. Starting with an initial vector x° ^ 0, and a fixed dimension

s > 1, construct the sequence \x } described above. Then

l i m\yi

Proof. From (12), the sequence \vι\ is a strictly decreasing sequence bounded

from below by λx . Hence there is a number v such that

242 W. KARUSH

lim V1 = V >

By (12) the smaller root v\ of the polynomial pι

2 (λ ) is not less than vι. Hence

pi{vi) = (vi - μ * ) (Vi - μi) - (t\f>0,

(t})a < (^"'-v'KM-v*),

since μι

0 = μ ( x ι ) = V1""1 [see (2) and (11)] . By (1) there is a constant M, inde-

pendent of i, such that

(17) (ίi) 2 < M(vi"1-vi) .

In particular,

ί ι —> 0 as i —> oo.

Recalling (13), put

Thus

Γ, - Σ 6JτΛ

From (14) and the definition of t\, we have

(*ϊ) 2 = ΓTTi = ( M ) a [ « ( λ i ) ] a + ••• + (bjo)2 [ p i ( λ r o ) l 2 .

Since the sum of squares on the right tends to 0, each term must do the same. But

Pi(λy) = (λy — μ^) - (λj — v1"1) —> (λy — ? ) . From the second equation of

(18), it follows that for some index / we have

v = , | 6 } | — » 1 , 6 } — > 0 for j• £ I .

(The last two conditions follow from the distinctness of the λy.)

We propose to show that I = 1. Suppose I l Then

α»


Jnl.JiίL . WL ^ oIJ-,1 I'll Ml

Using (12), we have

\ < λj < vi < ή 0 = 1 , 2 , •••, s - l ) .

It follows that p*. ( λ ) has the same sign at λ = λ j , λ l f v ι . Furthermore, s ince

by Lemma 3 this polynomial has only real roots , we have

\P}(\)\ > |pj(λ,)|.

Thus in formula (15) each term in braces for the coefficients a\ and αj is positive,

and each term for a\ is not smaller than the corresponding term for α j . Hence,

for all i, we have

| α l 1 | \a\\

By assumption, α£ = 1, k = 1,2, , r ° . We now have a contradiction to (19).

Thus I = 1.

Since a\ > 0 by (16), we have b\ > 0. Hence

6 j — > 1 , 6 j — > 0 f o r j φ \ .

The theorem now follows from the first equation of (18).

6. The main theorem. Before proving the principal result, Theorem 2, we

establish two lemmas.

LEMMA 8. Let 13 be an invariant subspace with lowest characteristic value

\χ having multiplicity one. Then for x ^ 0 in B, we have

( \ X <r 1 l£(*) l 2 , , s^ Xμ{x) ~ λi < T-T- * —:—~— whenever μ{x) < Λ.2 .\χ

Proof. (An alternative proof, applicable to normal matrices, is given by H.

Wielandt [4] .) Write x in the form (4) where y , y 9 , yι is a complete set of

orthonormal characteristic vectors in B. We let

* =x - , μ* = μ(x*) ,

244 W. KARUSH

and

ξ=i(x) ~Ax ~ μx , ξ* =ξ(x*) =Ax* - μ*x*

From (#*, yx) = 0, we obtain

(r,yi)=o.

From this and {ξ , x*) = 0, we obtain

From the definition of ζ , we have

£* = Λ* - αxλiyx ~μ*x

= ^ - (/i* ~μ)x + (/x* - λ ^ α i y ! .

Hence

0 = (^ , x) =~(μ* -μ)\x\2 + (/*• - λ i ) β ?

Also

0<(ξ*. ξ*) = {ξ\ ξ)= | ^ | 2 + ( / x * - λ 1 ) ( λ 1 - A t ) α ?

from the definition of ξ. Eliminating a\ from the preceding equation, we obtain

Since x* £ B and x* is orthogonal to yj, we have

μ* > k2 .

Hence, whenever μ < λ 2 , the inequality of Lemma 8 follows from the second

inequality above.

We shall eventually show that the sequence of lengths \xι\ converges. To do

this we shall require a bound on the ratio |p ι . (vι) | / τ * . . This is obtained in the

next lemma.

LEMMA 9. Suppose that for all i wte have s < rι\ Then there exists a constant

K, independent of i and /, such that for i sufficiently large we have

|pj(^)l <K(τ/)2 0 = 1 , 2 , •", s-1) .

Proof. By Theorem 1, we have μ(xι) — V1 —* λ1# Hence we may confine

ourselves to is so large that, say,

v ^ - λ , < ( l / 2 ) ( λ 2 - λ x ) .

Consider first / = 1. Apply the inequality of Lemma 8 with x = xι, B = (x°. We

find that

By (11), we have

and

Hence

(20)

as desired.

Let

D

1

λ.2 ~"" μ \xι)

Ip ί i

1λ2 - vι~

2<c — •••• ••

λ 2 — kx

t\)2,

( v -

The inequality (20) may be written R\ < K. We propose to show that for some

constant Kι , independent of i and , we have

(21) R) < K^RJ.J2 0 = 2 , 3 , •", s - 1 ) .

This, together with (20), will establish the lemma.

For the remainder of the proof we omit the superscript i. Writing py(λ) as a

product of linear factors, we obtain from (12) and (7) the result that

(22) \p. (y) \<K2\V-Vj\< K2{Vj ~ λj) .

2 4 6 W. KARUSH

In order to estimate the last difference we make use of the minimum characteristic

vector z relative to the subspace fly = (xi

9 Ax1, , A}~1 xι).

We have

μ(z) = Vj .

By (12) we may apply the inequality of Lemma 8 with x = z and B = fl°. Thus

where

ξ{z) =AZ~VJZ.

The vector ξ{z) is orthogonal to fly and lies in βy+i By (3) the vector is a scalar

multiple of ξj. To determine the scalar we use (8) and (2). We find that

Since (v* =) v < Vj < Vj-ι, the above coefficient of ξj does not exceed Rj-χ

(= i?l._j) in absolute value, Vj-ι being the least root of the polynomial. Also

U l 2 > I* I 2 , by (8). Thus

( 2 4 )

The combination of (22), (23), and (24) yields the desired inequality (21).

We turn to the main theorem. The theorem has an obvious counterpart for the

maximum characteristic vector.

THEOREM 2. Let A be a real symmetric operator on a real vector space of

dimension n. Given an initial vector x ψ 0 and a fixed dimension s (1 < s < n),

construct a sequence of vectors {x } as follows: let x1*1 be the unique minimum

characteristic vector relative to the subspace &s(xι) of the form xι + Ύ)1, with

(x\ T)1) = 0. Then xι converges to the minimum characteristic vector in flU°),

the smallest invariant subspace containing x°. Furthery the vector xι ι is given

by (10), and the least root of ps(M converges to λ t , provided (9) holds. (In the

event that condition (9) fails, the sequence \xl\ is eventually constant, as re-

marked in the last paragraph of §4.)

Proof, By Theorem 1, it is sufficient to show that the increasing sequence

\xι | 2 converges. It is an easy consequence of (10) that

l*i+T=UT Π

where

* » • ; .;.,

By a well-known theorem on infinite products, to prove the desired convergence it

is sufficient to verify that Σ ^ = o c converges. By Lemma 9, this requirement is

reduced to showing that each of the ser ies Σ^°=o ( τ . ) 2 converges. For j = 1,

this ser ies converges by (17). There is, a constant X t such that \AX\ < Kι \x\

Using this inequality and (2), we obtain

\η+1\<Hence we have

< K2+ ή

It follows that for all i we have

ή < 0 = 2 , 3 , ••', s-1) .

The convergence of the remaining series now follows from the convergence for

; = 1. This completes the proof.

REFERENCES

1. M. R. Hestenes and W. Karush, A method of gradients for the calculation of thecharacteristic roots and vectors of a real symmetric matrix. To appear in J. ResearchNat. Bur. Standards.

2. L. V. Kantorovitch, On an effective method of solving extremal problems for quad'ratic functionals, C.R. (Doklady) Acad. Sci. URSS (N.S.) 48 (1945), 455-460.

248 W. KARUSH

3. C. Lanczos, An iteration method for the solution of the eigenvalue problem oflinear differential and integral operators , J. Research Nat. Bur. Standards 45 (1950),255-282.

4. H.Wielandt, Ein Einschliessungssatz fur charakteristischeWurzeln normaler Matrizen,Arch. Math. 1 (1948-1949), 348-352.

UNIVERSITY OF CHICAGO AND

NATIONAL BUREAU OF STANDARDS, LOS ANGELES

ON THE NUMBER OF INTEGERS IN THE SUM OFTWO SETS OF POSITIVE INTEGERS

HENRY B. MANN

1. Introduction. Let A9 B9 be sets of nonnegative integers. We define

A + β = {a + b]a € Af b € B ' By A°9 BΌ

9 we shall denote the union of

A9 B9 and the number 0, by A(n) the number of positive α's that do not ex-

ceed n. We further put

g.l.b. = α ,n

g.l b.

If 1,2, , k - 1 G 4, A ζf 4, we further put

(4) g.l b. = cXi .π ^ n + 1

The real number α is called the density of A9 OLi the modified density9 and α

the asymptotic density of A. Densities of A9 B9 C9 will be denoted by the

corresponding Greek letters α, β, γ, .

Besicovitch [ l ] introduced (X*, and Erdos [2] OLχ.

The author [3] proved: //C = ° + B for B 3 1 and A0 + β° otherwise,

then for all n ^ C we have

(5) C(n) > α * n + β ( n ) .

It was also shown [3I that in (5), (X* cannot be replaced by (X.

Received November 13, 1950.Pacific /. Math. 1 (1951), 249-253.

249

250 HENRY B. MANN

It is the purpose of the present note to improve (5) to the relation

(6) C(n) >0Lιn +B(n) .

The proof of (6) requires only a modification of the proof of (5), but will be

given in full to make the present note self-sufficient.

The inequality (6) immediately yields

(7) 7 > α ! + /3

if C has infinitely many gaps.

Now (7) is sometimes better and sometimes not as good as ErdόV [2j ine-

quality

(8) y > α + β/2

for the case 0ί > β, B 3 1, C = A0 + J3°. (To establish (8) it is really suf-

ficient to assume that there is at least one b° such that b° 4- 1 C B ) However

(7) holds also for C = A0 + B if B 3 1, and for C - A0 + B° without any restriction

o n β .

2. Proof. We shall now give a proof of (6) for the case C — A0 + B, δ 3 1,

and then shall indicate the changes which have to be made if nothing is assumed

about B but if C = A0 + B° . By α, b, c, * # we shall denote unspecified

integers in A9 B9 C, .

Let τ%ι < n2 < be all the gaps in C. Put nr — n, n — n t = d^ for i < r.

If there is one e £[ B such that

(9) a + e + cίi = Πj ,

form all numbers e + dt for which

(10) α + e + d t = n 5 , ί < r , s < r .

Let Γ be the set of indices occurring in (10). Put B = \e + rfs5 se^.

It is not difficult to prove the following propositions.

PROPOSITION 1. The intersection β f l δ * is empty.

PROPOSITION 2. Γλe integer n is not of the form a + e + ds for any s.

Since (10) also implies

(10') α + e + ds = nt ,

ON THE NUMBER OF INTEGERS IN THE SUM OF TWO SETS OF POSITIVE INTEGERS 2 5 1

it follows that β* contains as many numbers as there are gaps in C which precede

n and which are not gaps in A + B U fi*. Hence we have the following result.

P R O P O S I T I O N 3. If B U B* = Bί9 A + Bt = Ci9 then

( I D Ct(n) -C(n) = B i ( π ) - f i ( π ) .

Thus we have proved the following lemma.

LEMMA. If there is at least one equation of the form a + b + c?, = ΠJ9 then

there exists a B\ 3 B such that C± = A + Bx does not contain n9 and such that

(12) d ( n ) ~C(n) = B i ( π ) - β ( n ) > 0 .

Now let C = A0 + β, B 3 l Clearly, n t > 1. The numbers smaller than n\

are either in B9 or of the form n^ ~" α, or of neither of these two sorts. Also

Tii ft β, since C 3 β. Hence we have

(13) Cta) = m - 1 >Λ(ni - 1 ) +β(m) .

Since δ 3 1, we must have ΛI —1 €j! /4, (Λ! ~ 1) >&. Thus, we obtain

(14) C(ni) >

We proceed by induction and assume (6) proved, when n is the /th gap, / < r.

We distinguish two cases.

Case 1: dΓ-i < n^ . Then

C 3 n i - cfΓ-i = α + 6

We now apply the lemma. Let n be the /th gap in Cχ Then j < r9 and we have, by

induction,

(15) d(n) >0Lιn

and, by the lemma,

U6) C r(n) ~C(n) = B

Subtracting (16) from (15), we obtain (6).

Case 2: c/r—]. > n p Now

n — nΓ-i — l >

252 HENRY B. MANN

Hence we have

A(n - nr-x - 1) > CLχ(n - n r - i )

The numbers between nr-γ and n are either of the form n ~~ α, or in B9 or of

neither of these two sorts. But n ££ B hence,

(17) n - n r - ! - 1 > A(n - nr-x - l) +B(n) - β ( n r - i )

> OL^n - nr-t) + β ( n ) - β ( n r - 1 ) .

By induction we have

(18) C(n r-X) = n Γ -i - (r - l ) > Ot^r-i +J5(n r- 1) .

Adding (17) and (18), we obtain (6).

From the proof it is evident that we may obtain the even stronger inequality

( 6 ' ) C(n) > B(n) min

To establish (6) for C = A0 + B° without the restriction B 3 1, we first

remark that in (13) the term A{nχ — 1) can be replaced by A(τiι). The cases to be

distinguished are dr-ι < n t and dΓ-χ > n\ . The proof of Case 1 is then word by

word the same when we replace B by B° and Bι by B J . In Case 2 we have

n — nr~ι ~ 1 > Πi > k ,

so that /4(τι — Λr-i ~1) > αt(w — flr-i); ^ e remainder of the argument remains

unchanged. For C = i4° + S° , we can obtain the even stronger inequality

(6") C(n) > B{n) mm

which again implies the even stronger result

C(n) > max

(n) -h + minΠi<n

ON THE NUMBER OF INTEGERS IN THE SUM OF TWO SETS OF POSITIVE INTEGERS 253

To establish (7), it is sufficient to show that for any set S we have

S(m) S(n)

m n

if m > n9 n €£ S, S(m) — S(n) — m ~~ n. However, this can easily be verified.

Thus if S has infinitely many gaps, then

- . . . -5 (« ) . S(n)cr = lim m i = lim m i .

m n$S n

It thus appears that in (7) we may replace β by

Bin) -lim inf - ^ > β .

n$C n

If C = A0 + B° , we may of course write

Ύ >max (α! + 3 , OC

REFERENCES

1. A. S. BesicoVitch, On the density of the sum of two sequences of integers, J.LondonMath. Soc. 10 (1935), 246-248.

2. P. Erdos, On the asymptotic density of the sum of two sequences, Ann. of Math. 43(1942), 65-68.

3. H. B. Mann, A proof of the fundamental theorem on the density of sums of sets ofpositive integers, Ann. of Math. 43 (1942), 523-527.

OHIO STATE UNIVERSITY

A THEOREM ON THE REPRESENTATION THEORYOF JORDAN ALGEBRAS

W. H. M I L L S

1. Introduction* Let / be a Jordan algebra over a field Φ of characteristic

neither 2 nor 3. Let a —» Sa be a (general) representation of /. If Ot is an alge-

braic element of /, then S^ is an algebraic element. The object of this paper is to

determine the polynomial identity* satisfied by S α . The polynomial obtained de-

pends only on the minimal polynomial of α and the characteristic of Φ It is the

minimal polynomial of S α if the associative algebra U generated by the Sa is the

universal associative algebra of / and if / is generated by (X.

2. Preliminaries* A (nonassociative) commutative algebra / over a field Φ is

called a Jordan algebra if

(1) (a2b)a = a2(ba)

holds for all α, b £ /. In this paper it will be assumed that the characteristic of

Φ is neither 2 nor 3.

It is well known that the Jordan algebra / is power associative;** that is,the

subalgebra generated by any single element a is associative. An immediate conse-

quence is that if f(x) is a polynomial with no constant term then f(a) is uniquely

defined.

Let Ra be the multiplicative mapping in /, a —> xa = ax9 determined by the

element α. From (1) it can be shown that we have

[fiα«δc~] + [RbRac] + [RcRab] = 0and

RaRiRc + RcRbRa + R(ac)b = RaRbc

for all α, b, c £ /, where [AB] denotes AB — BA. Since the characteristic of

Φ is not 3, either of these relations and the commutative law imply (1). Let

Received November 20, 1950.

•This problem was proposed by N. Jacobson.

**See, for example, Albert [ l ] .Pacific h Math. 1 (1951), 255-264.

255

256 W. H. MILLS

a —* S α be a linear mapping of / into an associative algebra U such that for all

α, b9 c C / we have

(2) [SaSbe] + [SbSvc] + [ScSab] = 0

and

(3) SaSbSc + ScSbSa + θ ( α c ) 6 = θ α 56 c + SbSac + ScSab .

Such a mapping is called a representation.

It has been shown* that there exists a representation a —> Sa of / into an

associative algebra V such that (a) U is generated by the elements Sa and (b) ifa > Ta is an arbitrary representation of / then Sa —> Ta defines a homo-

morphism of {/. In this case the algebra V is called the universal associative alge-

bra of /.

We shall now suppose that a —> Sa is an arbitrary representation of /, and (X

a fixed element of/. Let s(r) = Sar,A = s( l) , B = s(2). If we put a = b = c = α

in (2), we get ^ δ = BA. If we put α = b = α , c = αΓ~ 2 , r > 3, then (3) becomes

(4) s(r) =2As(r - l ) + s ( r - 2) B - Λ 2 s ( r - 2) - s ( r - 2 ) A2 .

We now see that.A and 5 generate a commutative subalgebra ί/α containing 5(r) for

all r . By the commutativity of ί/α, (4) becomes

(5) s ( r ) = 2 Λ s ( r - l ) + ( β ~ 2 Λ 2 ) s ( r - 2 ) .

We now adjoin to the commutative associative algebra £/α an element C commuting

with the elements of ί/α such that C 2 = B - 4 2 . We have the following result.

LEMMA 1. For all positive integers r, we have

s(r) = (l/2)(A +C) r + (1/2)(A - C y .

Proof. If r = 1, then

(1/2)(A+C)r

*For a general discussion of the theory of representations of a Jordan algebra and aproof of the existence of the universal associative algebra, see Jacobson [2]

A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 2 5 7

If r = 2, then

(l/2)(A+C)r + (1/2)(>1-C) r = 4 2 + C 2 = s ( 2 ) .

Now suppose that r > 3 and that Lemma 1 holds for r — 1 and r — 2. By direct

substitution it follows that A + C and 4 — C are roots of

x2 = 2A* + β - 2A2 ,

and therefore of

xr = 2Axr-1+ (B - 2A2) xr'-2 .r =

Hence,

U + C)r = 2A(A + C)r"ι+ {B - 2A2)(A + C) r"2

and

(A -• C) r = 2A (A - C ) Γ - χ + (β - 2A2 ) (A - C)Γ~ 2 .

Adding and dividing by 2, we have the desired result:

(1/2)(4 + C)r + (1/2)(A-C)r =2As(r - l) + (B ~2A2) s(r - 2) = «(r) .

An immediate consequence of Lemma 1 is that if g(x) is an arbitrary polynomial

with no constant term, then

(6) S g ( α ) = (1/2) g(A +C) + (1/2) g(A - C) .

Now suppose further that GC is an algebraic element of / and that f(x) is a

polynomial with no constant term, such that /(a) = 0. Then by (6) we have

(7) 0 = 2 S / ( α ) = /(A + C) + /(A - C) ,

0 = 2Sα/(α) = (A + C) f(A + C) + (A - C) f(A - C) .

The next step is to eliminate C from the system (7). To do this we need some

additional tools.

3 Theory of elimination. Let Ω be the splitting field of f{x) over the field $ .

Let P = Φ[χ], Q = P[y], P' = Ω[x], <?' = P'[y] be polynomial rings in

one and two variables over Φ and Ω, respectively. Then P and P' are principal

ideal rings. If qx and g2

a r e elements of Q, let (qi9 q2) be the ideal of Q generated

by qί and q2 , and let { x, r2} be a generator of the P-ideal (ql9 q2) Π P. Simi-

larly, if ^ and q2 are elements of Q1 , let ((^i, ^2)) ^ e ^ e ideal of Q' generated

2 5 8 W. H. MILLS

by qι and q2 . Furthermore, let U<7i> <72> > denote a generator of the P ' - ideal

((<7i* #2)) Π P ' . We note that {qi9 q2] and {{qi9 q2}] are determined up to unit

factors. The unit factors are nonzero elements of Φ and Ω respectively.

We shal l establ ish the following lemma.

LEMMA 2. If qι and q2 are elements of Q, then {qχ9 q2] = liqi, #25 5 UP t o

a unit factor.

Proof Let ωi9 ω2 , # , ωm be a basis of Ω over Φ. Then P ' = Σω^ P and

(?' — Σ ω j ^ . Therefore

and

( ( q ! , q 2 ) ) Π P ' =Σωi((ql9q2) Π P) = ( ( g i , g 2 ) Π P) P # = { q i , g 2 } P ' .

It follows that foi, g2} = {{g l f ςτ2Π

Let r and s be distinct elements of P' , and let m and n be positive integers.

We shall determine {{{y - r ) m , (y - s)n]}.

LEMMA 3. Let S(m9n) be that positive integer satisfying

S(m, n) < m + n — 1 ,

and

\n - 1 = 0 i f S(m,n) < N < m + n - 2 ,

where I ..I is the binomial coefficient considered as an integer in Φ. Then we have

U ( y - r ) . (y - . ) " ! ! = ( - r ) 5 ( » ' " > .

Proof. We note that S(m9n) depends only on m, n9 and the characteristic p of

Φ . If p = 0, or if p > m + rc ~ 1, then S(m9n) = m + rc — 1. In any case,

(8) m + n - 1 > S (m, n ) > n .

Replacing y by y + r, we may assume that r — 0, s f1 0. Formally, modulo ym

9

we have

μ=0

m-l

1 μ=0

n + μ - 1n - 1

= Σv=n

- 1n - 1

Therefore there exists a q £ Q' such that

(9) qy" + (y-s)"(-l) n £ ss(κ

It follows that

- s ) B Π | s s ( " ' B )

Put

. (y - s ) B Π | s

\ h a , (y ~ s Y \ \ = G ,

Then G and H are elements of P' . Furthermore, there exist qt and g2 \n Q' such

that the y-degree of q2 is less than m and such that qιym + <72(y ~~ 5 ) Γ l ~ C.

Hence

(10) g i ^ y α + g 2 t f (y - « ) n = GH = s

s ( * π ) .

Subtracting (9) from (10) and comparing terms not divisible by y m , we obtain

S(m,n)

(11) q2H=(-l)n

Comparing coefficients of y* 5^' 7 1 ' """ in (11), we get

HS(m,n) - I 1

n - 1 J

2 6 0 W. H. MILLS

which is a nonzero element of Φ. Therefore H is a unit element, and this es-

tablishes Lemma 3.

In the following we shall use l.c.m. (aΪ9 a2, * * * , an) for the least common

multiple of ax, α 2 , * * ' , an.

LEMMA 4. If ((qί9 q2)) Ξ> P' 9 then

Proof. P u t p t = {{qϊ9q3]} , p 2 = U ^ ^ Π , and p 3 = l.c.m. ( p ^ p 2) .

We note that((qlf q3)) DP' 2 (iqιq2* £3)) Π P ' , and therefore P l | {^1^2 > Π

Similarly, p 2 I U^i^2> ^3 I L and hence p 3 | U^i^2> ^ 3 ^ Now there exist D, £,

F, G, H, ΓinQ' such that

Therefore

Dqxq2 +

Hence there exist Kf L, M9 N in Q1 such that

Kqtq2 + Lq3 = p 3 g 2 and

Hence

(fflf + I/f)gig 2 + (fflV + I L ) g 3 = p 3 .

Therefore {{qιq29 qzlW Ps 9 a n ^ t n e proof of Lemma 4 is complete.

We shall now determine ID, El , where

0 = / ( * + y) + / ( * ~ y ) ,

E = (x + y) f{χ + y) + (x ~ y) f{χ ~ y) .

By Lemma 2, we have \D,E\ = U ^ . ^ H - Since

£ - (x - y)D = 2y/(χ + y) ,

we have

+ y ) \ } .


Put

lif{χ+y),f(χ-y)tt =Δ.

Let n be the degree of /(#). Choose F(γ) and G(γ) in Q1 , with y-degree less than

n, such that

F(y) f(* + y) + G(y) f(χ - y) = Δ.

Then F(y) and G(y) are completely determined. Now

F(-y) fix ~ y) + G(-y) f(x + y) = Δ.

Therefore we have F( —y) = G(y), from which it follows that F(0) = G(0), or

( F ( y ) - G ( y ) ) / ( * + y) + Giy) D = Δ .

Therefore U θ , y / U + y)}} | Δ . It is clear that Δ | {{D,yf(x + y)\}. Thus we

have

{ D , E \ = { { D . y f i x + y ) ] ] = Δ .

We must now determine

Let/( ie) = Π(Λ; - 0Li)ni, where the 0ίt are distinct elements ofΩ . Then

/(* + y) = Π(* + y - cci)"' , fix - y) = Uix - y - α; )"; .

If qι and q2 are two relatively prime factors of f{x + y), or of f(x "~y), then

((^rt, ^r2)) — P' Therefore we can apply Lemmas 3 and 4 to obtain

(12) \D,E] = Uf(χ+y), f(χ-y)}} = l.cm. (2χ - a* - aj)s^'nJ).

4. The equation for Sα . We shall establish the following result.

THEOREM. Let α be an algebraic element of J satisfying the equation

/(cc) = 0, where f(x) is a polynomial with no constant term. Let

fix) = Π ( * - C ί i ) Π i .

2 6 2 W. H. MILLS

where the 0L( are distinct elements of the splitting field Ω of f(x). Put

φ(x) = l .c.m.(* - (l/2)<Xi- (l/2)αy) 5 ( l | i "J> .

Then φiSa) = 0. Furthermore, if the algebra U generated by the Sa, a £ /, is

the universal associative algebra of J9 if fix) is the minimal polynomial of CL9and

if J is generated by Ct, then φix) is the minimal polynomial satisfied by S<χ.

Proof. As before, we let P = *>ix] 9 Q ~ P[y] be polynomial rings over $ in

one and two variables respectively, and put

D = / ( * + y) + f(χ ~ y)

and

E = (x + y) f{x + y) + (x - y) f{x - y) .

From (7) and (12) it follows that \jj{Sa) = 0. We must now show that φ(x) is the

minimal polynomial of Sα under the three given conditions. If we let (/(%)) be the

principal ideal of P generated by fix), then / is isomorphic to the quotient ring

P/ifix)) under the natural mapping g(θί) —» gix) + ifix))> Let V be the quotient

ring Q/iD,E)* We now consider the linear mapping

(13) g ( x ) _ * Γ g W = (l/2)g(* + y) + ( l/2)g(* -y) + (D,E)

of P into V. By the commutativity of V we have, for all g9 h9 j C P9

(14) [TgThj] + [ThTgj ] + [TjTgh] = 0 ,

since each of the three terms vanishes. Furthermore, by direct substitution we have

(15) 2TgThTj + Tghj = TgTh] + ThTgJ + TjTgh .

We now determine the kernel K of the mapping (13). By definition, gix) G ί if

and only if gix + y) + gix — y) C (D9E). Now

yf(x + y) = (1/2) £ - (1/2) (* - y) D C (D,E)

and

y/(* - y ) = (1/2)(x + y)D - (1/2) £ € (D,E) .

Let gr(%) be an arbitrary element of P. Then, for suitable hix9y) C (?> w e have

A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 263

+y) /(* + y) +g(*-y) /(* -y) = ?(*)£ +M*.y) y/(χ +y)

Therefore q(x)fix) C K for all q(x), and thus K 5 (/(*))• Suppose gθt) £ K9

gix) €£ (/(#)). We may suppose that the degree of gix) is less than n, the degree

of fix). Then gix -f y) + g(% — y) = hj) -\- h2E for suitable Ax and A2 in (?.

Since the degree of D is n and that of £ is n + 1, it follows that hγ — h2 — 0.

Therefore g(% + y) + gix ~~ y) is identically 0. This implies that gix) is identi-

cally zero, a contradiction; hence we have K = ifix)). It follows that

g(α)—> Γg(;t) = (l/2)g(x +y) + (l/2)g(x -y ) + (A*)

defines a single-valued linear mapping of / into V\ Furthermore, (14) and (15)

imply that this mapping is a representation, and from (12) it follows that Tx, the

image of OC, has \pix) — {D9 E] as its minimal polynomial. Now since U is the

universal associative algebra of /, the mapping Sg(oς) —> g(x) defines a homo-

morphism * of U into V. It follows that φix) is the minimal polynomial of S α . This

completes the proof.

We conclude by mentioning two simple consequences of the main theorem.

If /(*) =• xn , then ψix) = χs(n>n\ Now (8) yields S(n,n) < In - 1, and we

have the following result.

C O R O L L A R Y 1. // α71 = 0, then S&n~ι = 0.

Similarly, we obtain the following result.

C O R O L L A R Y 2. Let fi<x) = 0, where

Then A(Scχ) = 0, where

A(x)= Π (χ-(l/2)βμ-(l/2)βv).

18 In fact it can easily be shown that this mapping is an isomorphism of U onto V,

2 6 4 W. H. MILLS

Proof. Suppose

where the α, are distinct. Now by (8),

S(rii, Πj ) < Πi + Πj: — 1 <

and

A(χ) = Π ( χ - α i ) n ^ + l V 2 Π (* ~ ;i j>i

Therefore \p{x) | Λ(Λ ) , and the second corollary follows.

R E F E R E N C E S

1. A. A. Albert, A structure theory for Jordan algebras, Ann. of Math. 48 (1947),546-567.

2. N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math.Soc, scheduled to appear in vol. 70 (1951).

YALE UNIVERSITY

10

AN APPROACH TO SINGULAR HOMOLOGY THEORY

TlBOR R A D O

I N T R O D U C T I O N

0.1. Given a topological space X, we associate with X a complex R — R(X) as

follows. Let £00 denote Hubert space (that is, the space of all sequences r l 5 ,

rn, of real numbers such that the series r\ + + τ\ + converges,

with the usual definition of distance). For p > 0, let t>0, , vp be a sequence

of p + 1 points in £00 , which need not be linearly independent or distinct, and

let |ι>0, , Vp I denote the convex hull of these points. Finally, let T be a

continuous mapping from |t>0> * * > vp I in^° ^ Then the sequence vo, , Vp

jointly with T is a p-cell of the complex R, and will be denoted by(t>0># * #>

Vp, T) . The group Cp of (integral) p-chains in R is defined as the free Abelian

group with these p-cells as free generators. For p < 0, Cp is defined by Cp = 0

(that is, Cp consists then of a zero-element alone). The boundary operator 3p :

Cp —> Cp~-ι is defined by the conventional formula

i=0

for p > 1. For p < 0, Bp is defined as the trivial zero-homomorphism. Clearly

B3 = 0, and thus R = R (X) is a complex which is obviously closure-finite in the

sense of [ 4 ] . Accordingly, one can define cycles zp, boundaries bp, and so

forth, for R in the usual manner. The homology groups of R are defined by Hp =

, where Zp , Bp denote the group of p-cycles and p-boundaries respectivelyP/ ^P ' WUCJL"C ^p » upinR.

0.2. The complex /?, which was introduced and studied recently by the writer

[6] , differs from the various singular complexes used in previous literature first

in the use of Hubert space. The general practice is to consider continuous map-

pings T from rectilinear simplexes located in any Euclidean space. Instead, we

Received November 8, 1950.Pacific J. Math. 1 (1951), 265-290.

265

266 TIBOR RADO

use Hubert space in its capacity of infinite-dimensional Euclidean space, a pro-

cedure which may of course be adopted in all the various versions of singular

homology theory. The main departure from previous practice lies however in the

fact that no identifications are made in the chain groups Cp of R : two p-cells

(t>o ># " # >vp > T1 ) R f (t>2 * * * # >vp * T ")R a r e e q u a l i f a n d o n l y i f t h e y are identi-

cal, that is, if VQ = VQ, , Vp — υ'p, T' = T" * Thus the complex/? is of

enormous size as compared with previously used complexes. Let us note that

beyond the lack of identifications, R is further increased by the fact that the

points t>0, , Vp occurring in a p-cell (vo, , vp, T) are not required to be

linearly independent or distinct.

0.3. There arises the question of how the homology groups of R compare with

those arising in previous approaches to singular homology theory. In [6] , the

writer proved that the homology groups of R are isomorphic to those of the so-

called total singular complex 5 = S(X) introduced by Eilenberg [3] . Since this

result will be used in the sequel, we shall now give the precise statement of the

main theorem established in [6] . For each dimension p > 0, let us select a funda-

mental p-simplex, with (linearly independent) vertices d0, ?^p For our own

purposes, it is convenient to choose d0, dί9 d2,* as the points (1,0,0,0, •)>

(0,1,0,0, •)> (0,0,1,0, •), * in £Όo. Given then a sequence v0, , vp

of p -f 1 points in £00 > which need not be linearly independent or distinct, there

exists a unique linear map CC: | dQ, , dp \ —* | v0, , vp \ such that <x(dι)

— V(, i = 0, , p. This linear map is denoted by [v0, , vp] . The total singu-

lar complex S = S(X) of Eilenberg [3] may now be described as follows. For

p > 0, a p-cell of S is an aggregate (do, , dp, T) , where T is a continuous

mapping from \do, , dp | into X. The group Cp of (integral) p-chains of S is

then the free Abelian group with these p-cells as free generators. For p < 0, one

sets Cp = 0. The boundary operator Bp : Cp —* Cp-.χ is defined by

for p > 1. For p < 0, 'dp is the trivial zero-homomorphism. The homology groups of

S will be denoted by Hp We have then obvious homomorphisms

p *-* p f{^p 1 'p <-* p f ^ p

defined as follows for p > 0:

AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 6 7

τp{dor~,dp,T)s= (do,'~,dp,T)R ,

<rp(vo,'-,Vp,T)R= (do,' ',dp,T[vo,- ,vp])s .

For p < 0, τp and σp are defined as the trivial zero-homomorphisms. Unfortunately,

Tp is not a chain-mapping. On the other hand, σp is easily seen to be a chain-

mapping, and hence it induces homomorphisms σ*p : Hp —> Hp. The main result

of [6] is contained in the following statement.

THEOREM. The homomorphism σ*p : Hp —> Hp is an isomorphism onto, for

every dimension p.

Since singular homology theory is sometimes thought of only in relation to

triangulable spaces, it may be appropriate to note that the preceding theorem is

valid for general topological spaces. In particular, the space need not be arc-wise

connected.

0.4 In view of the preceding theorem, the complex R appears as an appropriate

tool in constructing singular homology theory. It is of interest to note that the

various complexes used in previous approaches to singular homology theory may

be derived from the complex R by a combination of the following two types of

reduction.

(i) The chain groups Cp of R are replaced by certain subgroups Γp. For

example, one may select Γp as the group generated by those p-cells (vo, ,

vp, T) for which the points vo, , vp are linearly independent. Another sig-

nificant choice may be based upon the concept of a minimal complex studied by

Eilenberg and Zilber [3]

(ii) One selects in Cp9 for each p, a certain subgroup Gp, and one replaces

Cp by the factor group Cp/Gp. From the computational point of view, this amounts

to an identification of elements of Cp which are contained in the same coset

relative to Gp. For brevity, we shall refer to this type of process as an identi-

fication scheme.

In the present paper, we shall study the effect of the various identification

schemes, occurring in previous theories, upon the homology structure of the com-

plex R, It is easy to see that these identification schemes may be reduced to

three basic types. Our result is that one may apply these basic identification

schemes in any desired combination without changing the homology structure of R

(see Theorem 1 in §4.7). As a matter of fact, we obtain an identification scheme

which appears stronger than those previously used (see Theorem 2 in 4.7 and see

268 TIBOR RADO

§5). This leads to some interesting questions, formulated in §6, which seem to

deserve further study.

0.5. It should be noted that the complex R is semisimplicial in the sense

of L3J > and therefore can be used to construct a complete homology and coho-

mology theory.

0.6. Previous relevant literature, as well as further problems arising in this

line of thought, will be discussed in §6 when convenient terminology will be

available. The writer wishes to express his appreciation of the courtesy extended

by S. Eilenberg and N. Steenrod who made available to him the manuscript of their

yet unpublished book [2] . Both technically and conceptually, the study of that

book proved most valuable in preparing the present paper.

1. I D E N T I F I C A T I O N S IN MAYER C O M P L E X E S

1.1. A Mayer complex M is a collection of Abelian groups Cp, where the

integer p ranges from — °° to + °°, together with homomorphisms

dp Cp > Cp-i ,

such that 3 p - ! Bp — 0. Cycles and boundaries are defined in the usual manner.

The homology groups Hp of M are defined by Hp = Zp/Bp , where Zp , Bp are the

groups of p-cycles and p-boundaries respectively. If M, W are Mayer complexes,

then a set of homomorphisms

fp : Cp >C'p

is termed a chain mapping if ^ή/p = fp-ι ^p > where primes refer to the complex

M1

For clarity, we shall write Cp, 3 p , Hp, and so on, to identify the complex

under consideration. In particular, a p-chain of M (that is, an element of C p) will

be denoted by symbols like cp , dp, and so forth.

1.2. We shall now describe the general process of identification in a Mayer

complex M. Let [Gp] be a collection of Abelian groups such that Gp C Cp and

(1) 3p Gp C Gp-X .

Explicitly: if c* € Gp , then d$c$ C Gp-X. Set Cj? = C$/Gp. Thus, the

elements of C™ are cosets relative to Gp . The general element of C ^ is of the

form {cp}, where this symbol denotes the coset containing the element cp of

Cp. In view of (1), we can then define homomorphisms

by the formula 3™ [cp] = [dpCp]. Clearly ~d™-x <3™ = 0. Accordingly, the system

of factor groups {C!?}9 jointly with the homomorphisms 3 p , constitutes a Mayer

complex m. We shall say that m is obtained by identification, with respect to the

system {Gp}9 from U. The system \Gp\9 satisfying (l),will be termed an identifier

for M We have then natural homomorphisms

TTp i C p * C p

defined by πp Cp = {cp]. Clearly

Thus Up is a chain mapping, and hence induces homomorphisms

defined as follows. If zp is a cycle in M9 then we let [2^ ]j/ denote the homology

class containing Zp . The symbol [z i$ j m is defined similarly. Then Tί^p is given

by

If rr^p is an isomorphism onto for every p , then we shall say that the identifier

{Gp} is unessential. Thus the process of identification with respect to an unes-

sential identifier does not change the homology structure of the complex.

1.3. We shall state presently a convenient condition for the unessential char-

acter of an identifier {Gp}. Let us observe that the condition (1) in 1.2 means

that the homomorphisms Bp, cut down to the subgroups Gp , may be used to turn

the system \Gp} into a Mayer complex which we call G. The complex m9 defined

in 1.2, appears then as merely the complex M mod G in the sense of the general

relative homology theory of Mayer complexes. From this general theory, the condi-

tion for 77*p to be an isomorphism onto, for all p, is well known: it is necessary

and sufficient that all the homology groups of G be trivial. For convenient appli-

cation, we shall now state this condition explicitly.

The condition (ί/). We shall say that the identifier {Gp} satisfies the condition

270 TIBOR RADO

([/) if the following holds: if zίf is a cycle in M such that zp G Gp then there

exists a (p + l)-chain Cp + X G Gp+t such that 3p+! Cp+1 = zp .

We have then the following criterion.

CRITERION FOR UNESSENTIAL IDENTIFIERS. An identifier \Gp] is unes-

sential if and only if it satisfies condition (ί/).

Since the elements of Gp represent those elements of Cp which are, in a sense,

discarded as we pass from the complex M to the complex m, the criterion may be

also worded as follows: discarded cycles should bound discarded chains. In a

special case, this criterion was used by Tucker [ β ] . As mentioned above, the

general criterion is merely a re-wording of a well-known theorem in the relative

homology theory of Mayer complexes (for a comprehensive presentation, see

Eilenberg and Steenrod [2]). For the convenience of the reader, we shall now

outline a direct proof of the criterion.

1.4. Assume first that the identifier {Gp\ is unessential. Take a cycle

(1) z^CGp.

Then 7Tpzp = \zp] = 0, and hence 77 * p [zp']nf = [τrpzp]m = 0. Since π * is an

isomorphism onto, it follows that zp bounds in M:

(2) 4 = 3? + 1 c*+ 1 .

Application of ττp yields, in view of (l), the equation

0 = 77p Zp = TTp'dp+i C p + ι = ^ + l 7 T + C

Thus Ttp+γ cp + t is a cycle of the complex m. Since 77* is an isomorphism onto, we

have therefore a cycle Zp+X such that 77p+12:p+1 differs from the cycle 77p+1Cp+1

only by a boundary. Thus we can write

Ήp+l Zp+l = πp+l c

Now c^+2 is of the form {cp+ 2 i = 77p+2c^+2 Making this substitution, we obtain

z p + l = πp+i c p + l +<^p + 277p + 2 c p + 2 = ^p+lfcp + l + ^ + 2 C

Hence

(3) ^p+lC^p+l ~~ zp + l + ^p + 2 c p+2) = 0

Now let us consider the (p -f l)-chain

°ί+l = c + i "" z

By (3) we have dp+ι G G p + 1 , while from (2) we have Zp = B "+i </"+! Thus (1)

is seen to imply that zp bounds a chain contained in Gp+ι. In other words, con-

dition (U) holds.

1.5. Assume now, conversely, that condition (ί/) holds. We have to show

that τr*p is an isomorphism onto for every p.

(i) Suppose we have

(1) 77.p[zp]* = 0

for a certain cycle Zp. The assumption means that TTpZp bounds some chain

c™+t. Since c^+ t is of the form {.Op*^ = 77p+i Cp+i > we have

πpZp = 3g + i77jp+1Cp + 1 =

and hence

πp{z$ -3p+iCp+i) = 0 .

Thus the cycle

(2) Zl= z^-B* + 1 c^ + 1

is contained in Gp. Since condition (ί/) is now assumed, it follows that Zp is of

the form

(3) Z £ = 3 j + 1 d p + i

From (2) and (3) it follows that

z = 3 + i ( c + + ίi +

Thus (1) implies that Zp bounds in M, and hence π*p is an isomorphism into,

(ii) Assign now an element [-z£*]m of H™. Now z™ is of the form

(1) z»=\4\=ττp4.

Since Zp is a cycle, we have

0 = BjJzjS = 3j}7Γpc£ = πp.jBJί cM

p .

272 TΓBOR RADO

Hence

Thus Bp Cp is a cycle contained in Gp-i Since condition (£/) is now assumed, we

(4€GP).

p p

have a chain dp such that

Thus Cp — dp is a cycle:

Now we calculate

rM _ ύί _ #p p — p

= \jτpcMp - πpdp]τpcp - πpdp]Λ .

By (1), Ήpc$ = z™, and by (2), π p ^ = 0. Thus finally

π pίzp]* = Op]. .

Thus 77+p is onto, and the proof of the criterion is complete.

1.6. In marked contrast to the general character of the preceding discussion,

the unessential identifiers actually employed in the sequel are of a very special

and restricted type. There arises the question whether there are general con-

structions yielding unessential identifiers in Mayer complexes. The following

comments may be of interest from this point of view. Let M, L be Mayer complexes

and let

(1) fp: CMp-*Clp

be a chain-mapping such that the induced homomorphisms /*p : Hp —» Hp are

isomorphisms onto. In symbols:

(2) f,p : U"p « HL

p .

Let Np denote the nucleus of the homomorphism (1). Since fp is a chain-mapping,

it is immediate that the system [Np] is an identifier.

In view of the strong assumption (2) one may be tempted to conjecture that

{Np} is unessential. The following simple example shows that this is not the

case, even under extremely special and favorable circumstances. Let M be a finite

simplicial complex described abstractly as follows. The group C^ of (integral)


2-chains of M is the free Abelian group with a single generator ί. The 1-chain

group C t is the free Abelian group with four generators si9 s2 9 s3, s 4 . The

0-chain group Co is generated by α, b, c, d, e. For p ^ 0, 1, 2, the p-chain group

Cp reduces to a zero-element. The boundary relations are as follows:

Bt = s 1 - f s 2 ~ l ~ $ 3 , Bsj = c — 6, Bs2 = α ~ - c , B s 3 = 6 -~ α, B s 4 = e — cί,

Bα = B6 = Be = Bcf = Be = 0 .

We define first homomorphisms /p : Cp —* C 5 as follows:

/2* = 0, fχS1 = / l S 2 = / l S 3 = 0, fιS4 = S! + 5 2 + S 3 ,

/o α = /o 6 = /o c = α, fod = foe = d .

For p ^ 0, 1, 2, of course /p is the trivial zero-homomorphism. Next we define

homomorphisms Dp : Cp —• ^ p + i a s follows:

ΰ o α = 0, D o f c = ~ " 5 3 , Do^ = s 2 , D o ^ = 0, Z)o e = — s 4 ,

For p ψ 0, 1, of course Dp is the trivial zero-homomorphism. One verifies readily

the following facts.

(i) fp is a chain-mapping.

(ii) BDp7p + Dp-iByp = / p γ£ - γ £ , for every p-chain y$ of Λf. Thus

(iii) Let /Vp be the nucleus of Λ,, and let m be the complex obtained from M

by using the identifier {/Vp},in the sense of 1.2. Then the l-dimensional homology

group Hψ of m is infinite cyclic.

(iv) The l-dimensional homology group //f of M is trivial (consists of zero

alone).

Thus M and m have different homology structures, and hence {Np} is certainly

not unessential. And yet, in view of (i), (ii), the induced homomorphisms f*p :

Hp —» tip are isomorphisms onto. In other words, a very plausible method to

obtain unessential identifiers fails even under very special and favorable con-

ditions.

1.7. In dealing with additively written Abelian groups, we shall use certain

familiar conventions. Thus we shall writ© G — 0 to state that the Abelian group G

274 TIBOR RADO

is trivial (consists of a zero-element alone). If A ι, , An are subgroups of G,

then A i + + An will denote the smallest subgroup containing A t , , A2

2. T H E AUXILIARY C O M P L E X K

2.1. The auxiliary complex K, which played an important role in [6] already,

is merely the "formal complex," in the sense of [2] , of £oo taken as a point set.

The complex K is defined as follows. For p > 0, a p-cell of K is a sequence

( υ o , , Vp) of points of E<χ3 which are not required to be linearly independent

or distinct. Two p-cells (v0 , , vp), (wo, * , wp) are considered as equal if

and only if v^ — w^ i = 0, , p. These p-cells are taken as a base for a free

Abelian group, to be denoted by Cp, the group of (finite) p-chains of K. For p < 0,

one defines Cp = 0. For p > 1, the boundary operator

3p : Cp —>Cp-1

is defined by the formula

Clearly 9 3 = 0 . For p < 0, <3p is of course defined as the trivial zero homo-

morphism.

Let {v0 , , Vp) be a p-cell of Kl Treating the points of £Όo a s vectors in the

usual manner, we describe the barycenter b — b(v0,* * *9Vp) of the points v0,

• , Vp by the formula

v0 + ••• + vp

b =p + l

2.2. The following homomorphisms will be used,

(i) The homomorphism 'dp : Cp —> ^n-i > already defined,

(ii) In terms of any assigned point v of £oo, one defines the cone homo-

morphism

hυ

p: Cp->Cp+1 (p > 0 )

by the formula

, • " , vp,v).

AN APPROACH TO SINGULAR HOMOLOGY THEORY 275

For p < 0, hp is the trivial zero homomorphism.

(iii) The barycentric homomorphism

βp Cp^Cp

is defined as follows. For p < 0, βp is the trivial zero homomorphism. For p = 0,

/30 = 1, the identity. For p > 1, βp is defined recursively by the formula

where b is the barycenter of the points v0 , , v γ.

(iv) The barycentric homotopy operator

Pp Cp > Cp+ι

is defined as follows. For p < 0, Pp(v0 , , vp) = 0. For p > 1, pp is defined

recursively by the formula

PP(v0 , •••, vp) = hhp(βp - 1 - p p - ! 3p)(v 0 , # , vp) ,

where b is the barycenter of the points (v0 , , vp).

(v) For p > 1, 0 < j y will be referred to as a transposition. Thus "transposition"

means here a transposition of adjacent elements. According to the definition of

equality for p-cells (see 2.1), we have tpfj(v0, , vp) = (v0 , , vp) if and

only if VJ = Vj+χ.

2.3. The following identities hold among these various homomorphisms

(i) V i hP + Λ £ - i 3 p = l (p > 1 ) .

(ii) -dpβp = ySp-j 3p ,

(iii) ^p + i^p + pp-i ^p = /3p ~ 1 ,

(iv) βptPtj=~βp ( p > l , 0 < j < p - l ) .

2.4. If (f0 , , Vp) is a p-cell of K, then j v 0 , , vp \ denotes the convex

hull of the points v0 , , vp (that is, the smallest convex set containing these

276 TIBOR RADO

points). If cp is a p-chain of K, and A is a convex set in Eoo » then the inclusion

Cp C /I is defined to mean that cp can be written in the form

CP = Σ fej(vo.j , ••*, Vp. j ) ,

where the coefficients kj are of course integers, so that |t>o,/> * * *> vp,j I

/ = 1, , n. One has then the following inclusions:

(i) ^P(vo , ' , t>p) C | v 0 , # # # , vpl ,

(") βp(v0 , , v p ) C l^o , ••*, v p | ,

(iii) Pp(vo .*••• v p ) C | v o > , V p L

As a consequence, an inclusion cp (Z A implies that ΉpCp C A, βpcp C A,

PpCp C A, tpjCp C /I. It is understood that the zero chain cp — 0 is agreed

to satisfy the inclusion cp C A for every convex set A.

2.5. For p > 1, an elementary t-chain in K is defined as a p-chain cp which

can be written in the form (see 2.2 (v))

cp = (vo > ###» vp) + tP,j(vo » # Ί *>p)

LEMMA. Given an elementary t-chain

Cp = (t>0, ", Vp) + t p , ; ( t > 0 , * , Vp) ( P > 1 ) t

ίAe following statements hold:

(i) //p = 1, £λeτι 3pCp ~ 0. //p > 1, ίAe^ 3pCp is α linear combination {with

integral coefficients) of elementary t chains d \ v0 , , Vp |

(ii) / 3 p C p = 0;

(iii) PpCp is a linear combination {with integral coefficients) of elementary

t-chains C | v0 , , vp \ .

Proof. The assertion (ii) is an immediate consequence of 2.4 (iv). The as-

sertions (i) and (iii) are readily verified for p = 1. Hence we can assume that

P > 1 .

Proof of (i) for p > 1. Le*t us note that tpfj{vQ, , vp) is of the form {w0 ,

• , Wp), where V( = «;; for j ^ /, / + 1, and ι?y = wy+i » t>y+i = w y. Now we have


Pp cp ~ 2* v — J- J l Λ v o > ι v i > » v p y ' V^o > i w i i* * iwp) 1

t=O

For i φ j9 j + 1, the quantity in square brackets is clearly an elementary ί-chain

^ I vo 9 * * * 9 vp I 0 n t n e other hand, the terms corresponding to i = / and

i = / + 1 cancel. Thus (i) follows.

Proof of (iii) for p > 1. Since (iii) is verified directly for p = 1, we proceed

by induction. Assume (iii) to hold for p — 1, where p > 2. Let us write again

ίp,/ = (^o ># * * 9 wp) Clearly, the points v0, , vp and the points w0, , Wp

have the same barycenter ό. Hence we have (see 2.2 (iv))

Pp {v0 , , vp)= hp [βp (v0 , , Vp) - (t;0 , , Vp)

= A

In view of (ii), addition yields

(1) Ppcp = hp( ~cp - Pp-i'dpCp) .

Now, by (i), 3pCp is a linear combination (with integral coefficients) of elementary

ί-chains C (v0 , , vp \. Hence, by the inductive assumption, the same holds

for yOp-i ^pCp9 and hence also for the quantity in parentheses in (1), and finally

for pp cp itself, since b C | v0 , , vp | .

2.6. For p > 1, an elementary d-chain in K is defined as a p-cell (ι;0 , , vp)

such that VJ = vy+i for some /.

LEMMA. // Cp = (vo, , Vp) is an elementary d-chain, then the following

statements hold.

(i) If p — 1, ίAew ^pCp = 0. //p > 1, ίAew 3pCp is a linear combination (with

integral coefficients) of elementary d-chains C | v0 , , Vp \.

(ii) βpcp = Q.

(iii) PpCp is a linear combination (with integral coefficients) of elementary

d-chains C | vQ , , vp | .

The proof is entirely analogous to that in 2.5, except that (ii) requires an

additional remark. We have VJ = Vj+t for some j by assumption. For this same j,

278 TIBOR RADO

we have then the relation

Hence we have also

βp tp9j(vOt 'fVp) = βp

On the other hand, 2.3 (iv) yields

Hence 2βp{v0, , v p) = 0. Since βp(v0, , v p) is an element of the free

Abelian group Cp , it follows that βp (v0 , , vp) ~ 0.

3. T H E C O M P L E X R = R(X)

3.1. In working with the complex R (see 0.1), the following device (introduced

by Eilenberg and Steenrod in [2] in connection with the complex S; see 0.3) is

useful. Let A be a convex subset of £Όo , and let Cp denote the subgroup of Cp

(see 2.1) generated by those p-cells (v0 , , vp) of the complex K which satisfy

the inclusion (v0 , , vp) C A (see 2.4). For p < 0, we define Cp = 0 (see

1.7). Let T : A —> I be a continuous mapping. We can define then homomor-

phisms

Γ # pk v pRp . L*p ' Li p

by the formula

7p(vo, ,vP) = (vo,~',Vp,T)R ( p > 0, (t/0, ,fp) C C^) .

For p < 0, Tp is the trivial zero-homomorphism. For Cp ζL Cp, it will be con-

venient to use the symbol (cp, T) to denote TpCp. Among the simple and obvious

rules of computation for the symbol (cp ,T) , we mention the formula

In terms of the preceding notations, we define now homomorphisms

Hp ' ^ p * v p 9

^.R . r* R v n RHp ^ p f ^ p + 1

by the formulas

βp(vo,'~,vp,T)R= (βP(vQ,~ ,vp),T)R, ( p > 0 ) ,

Since βp(v0, , vp) C | v0 , , vp | , pp(t>0 , , vp) C \vo, , vp\ hy

2.4, the homomorphisms βp, pp are well defined. For p < 0, /3p and pp are

defined as the trivial zero homomorphisms. In terms of the homomorphisms tp ,

defined in 2.2, we define

by means of the formula

tpj(vo,'",vp,T)R= (tPfj{v0, ~,vp),T)R .

We have then the following identities (see [6]) :

(1) 3«/3« = $-,3";

(2) 3 « + 1 / 0 « + p*^** = βR

p-l,

where 1 denotes the identity transformation in Cp; furthermore (see 0.3)

(3) # t * , , =-/3* ( 0 < J 1, we define an elementary t-chain in R as a chain of the form

(ι»o, " . vp, Tf + (tPfj(v0 ,'• >, vp), T)R (see 2.2). The subgroup of C* gener-

ated by the elementary ί-chains will be denoted by Tp . For p < 0, we define

T* = 0 .

280 TIBOR RADO

LEMMA. If C$ € Γp

Λ, then

(i) ^PcR

pCT^l

(ϋ) / 3 * c * = 0 >

(in) P p C j ? C ^

Proof. Clearly, it is sufficient to consider the case where cp is an elementary

ί-chain:

4= (vo,'",vp,T)R + (tPlj(v0,'",vp),T)R

= ((«o, •• ,vp) + tPtj(v0, ",vp),T)R .

Then we have

tf 4= (Bp[(iΌ. ,Wp) + tp,j(vo, ",vp)],T)κ.

By 2.5 (i), 'dp [(v0, , vp) -f tptj{v0 , , ι θ ] is either zero or else a linear

combination, with integral coefficients, of (p — l)-chains of the form (wo, ,

Wp-ι) + ^-1,^(^0 > * * *> Wp-i)» a^l c l^o ># * #» v p I > a n ( l thus (i) is obvious.

In a similar manner, (ii) and (iii) follow from 2.5 (ii) and 2.5 (iii).

3.3. For p > 1, we define an elementary d-chain in R as a p-cell (v0, * ,

vp, T) such that t>y = Vj+χ for some y, 0 < j < p — 1. The subgroup of Cp gener-

ated by the elementary d-chains is denoted by Dp . For p < 0, we define Dp = 0.

L E M M A , //cp G Dp, then

(ϋ) ^ c/J = 0 f

These statements are immediate consequences of 2.6 (i), 2.6 (ii), 2.6 (iii).

3.4 Given a p-cell (v0, , vp, T)R, take a sequence tυ0, , wp of p + 1

linearly independent points in E& . Then we have a linear mapping Cί: |w;0 , ,

Wp I —» I VQ , , Vp I such that &(wi) — v, , i = 0, , p. Then the p-chain

(1) 4= (vo,~',vp,T)R- {wQ," ,w

will be termed an elementary a-chain. The subgroup of Cp generated by the ele-

mentary a-chains will be denoted by Ap. For p < 0, we define Ap = 0.

LEMMA, C* G Ap if and only ifσpcp = 0 (see 0.3).

Proof. Assume Cp C Ap, Then c ί is a linear combination of chains of the

form (1), and hence it is sufficient to show that crpCp = 0 for the chain (1). Now

we have (see 0.3)

o-C = (do "dTlvo9mΛV])S ~ (rfo rf

Clearly [v0, , vp] = a [w0, , wp], and thus σpCp = 0.

Assume next that CpCp = (

can be written as a (finite) sum

Assume next that CpCp = 0. Then we also have TpσpCp — 0. The chain Cp

(2) 4= Σ njivo.j.—.Vp.j.Tj)*,j

where the coefficients nj are integers. We have then

(3) 0=T p <r p c*= ]Γ nj(dormm,dp,Tj[vo.j,"m,Vp,j'])*

j

Subtracting (3) from (2), we see that Cp appears as a linear combination of ele-

mentary a-chains, and thus cp £ Ap. If p < 0, then the lemma is of course

obvious.

3.5. LEMMA. If C* C A*, then

(i) δ£c£ CΛp-i

(ϋ) fiξc* CA«,

(ϋi) p$c* CA$+X.

These statements are immediate consequences of the identities (6), (7), (8) in

3.1, in connection with the lemma in 3.4. For example, to prove (iii), we note that

by (8) in 3.1, we have

(1) OΓp+i/ pcj} = σp+tpβrpσpc^ 0 ,

since Cp G Ap, and hence CpCp = 0 by 3.4. Also by 3.4, the relation (1) implies

282 TIBOR RADO

that pRc$

3.6. Let us observe that the chain groups Cp, Cp are free Abelian groups by

their very definition (see 0.3) and hence they do not contain elements of finite

order.

4. UNESSENTIAL IDENTIFICATIONS IN R=R(X)

4.1. LEMMA. Let {Gp} be an identifier for R (see 1.2, 0.1) such that the

following conditions hold:

(i) c*£ Gp implies that β* c$ = 0

(ii) cjj £ Gp implies that p$Cp£Gp + x .

Then {Gp} is unessential (see 1.2).

Proof. We shall verify that {Gp} satisfies condition (V) of 1.3. Take a cycle

Zp G Gp. In view of (i) and (ii), the homotopy identity

itf4 = βU - 4a) H+XPUI +

yields the relation

Thus z p is the boundary of the (p + l)-chain Pp Zp £ Gp+χ , and condition (U)

is established. By the criterion in 1.3, it follows that {Gp} is unessential.

4.2. LEMMA. Let {Gp} be an identifier for R, such that the following con-

ditions hold:

(i) Gp D A £ (see 3.4);

(ii) Cp CGp implies that crpβpCp = 0 (see 0.3);

(iii) Cp£Gp implies that ppCp CGp+i

Then {Gp} is unessential.

Proof. Again, we verify that {Gp} satisfies condition (ί/). Let us take a cycle

Zp £ Gp; we have to show that it is the boundary of some chain in Gp+i. We

note that

is a cycle, and that by (ii) we have

CΓpil =σpβ«zR

p = 0 ,

since Zp G Gp. Since σ*p : Hp —> Hp is an isomorphism onto (see 0.3), it

follows that ζp bounds:

(2) ζ j = 3 ? + i 7 p + i .

Applying σp on the left, we get (see 0.3)

0 = σ > ζ * = σ p 3 * + 1 y * + 1 = ^ ^

Thus σp+ιγp+ί is a cycle:

(3) σ+iΎ+i = ^

Since cr+ is an isomorphism onto (see 0.3), there exists a cycle Zp+i such thatzρ+ι a f ld CΓp+i Zp+i differ only in a boundary:

| 1 p + 1 £ 1 - f

- Zp + l ~ ^p + 2 ^ + 2 C + 2 ) = 0 .

Since 3p+2 = crp+1 3p+2 Tp+2 , the relations (3) and (4) yield

(5) <7>+l(7p

On setting

< 6> 4 + 1 = 7 p + l - ^p+l -

we see that the relations (5), (1), (2), (6) yield

(7) cr p + 1 d j + 1 = 0 ,

(8) βR

pz$ = 3 j ϊ + i ^ + i .

From the homotopy identity 4.1 (1) and from ,(8) we infer now that

(9) z =

By (7), (i), and 3.4, we have d*+ι G G p + 1 . Since p^z^ G Gp+i by (iii), it

follows from (9) that zp is the boundary of a chain in Gp+ι, and the proof of the

2 8 4 TBBOR RADO

lemma is complete,

4.3. LEMMA. Let \GΛ be an identifier for R which satisfies the assumptions

of the lemma in 4.1. For each p, let Gp denote the division-hull of Gp. Then \Gp\

is again an identifier (see 1.2) which satisfies the assumptions of Lemma 4.1.

Proof. Take a chain Cp C Gp. Then there exists an integer n φ 0, such that

ncp £ Gp and hence (since {Gp} satisfies the assumptions of Lemma 4.1)

(1) nβ» 4=0,

(2) np«4CGp+1.

C £ G+ Since βcBy the definition of Gp+i, (2) implies that Pp Cp £ Gp+i Since βpcp is an

element of the free Abelian group Cp (see 3.6), (1) implies that βpCp = 0.

4.4. LEMMA. Let \Gp\ be an identifier for R which satisfies the assumptions

of Lemma 4.2. Then {Gp} is again an identifier which satisfies the assumptions

of the same lemma.

The proof is the same as in 4.3, except that one uses now the fact that σpβp cp

is an element of the free Abelian group Cp ^see 3.6).

4.5. LEMMA. Let {G^}, ,{Gίn)} be identifiers for R, satisfying

the assumptions of Lemma 4.1. Then {Gp + + Gp } is again an identifier

which satisfies the assumptions of Lemma 4.1.

The proof is obvious.

4.6. LEMMA. Let Ω' be a collection (perhaps empty) of identifiers for /?,

each of which satisfies the assumptions of Lemma 4.1. Let Ω" be a nonempty

collection of identifiers for /?, each of which satisfies the assumptions of Lemma

4.2. For each p, let Gp denote the smallest subgroup of Cp containing the groups,

with the same subscript p, of the identifiers contained in Ω' and Ω". Then {Gp}

is an identifier satisfying the assumptions of Lemma 4.2.

The proof is obvious.

4.7. The preceding lemmas, combined with the results of §3, yield a number

of unessential identifiers for R. In the following two theorems, the symbols Ap ,

Dp, Tp have the meanings explained in §3.


T H E O R E M 1. Each one of the systems [A*], {D* } , {T*}9 {A* + D ^ j ,

\Ap + T*}, {D* + Tp], {A* + D* + T*} is an unessential identifier for R

(see 1.2).

THEOREM 2. If ΓJJ denotes the division-hull of the group Γ* = A* + θ£ +7]?,

then {Γp 5 is an unessential identifier for R,

Proof* By 3.5 and 3.4, the system \Ap } is an identifier satisfying the as-

sumptions of Lemma 4.2. Similarly, the systems {Dp }, {Tp} are identifiers

satisfying the assumptions of Lemma 4.1, by 3.2 and 3.3 respectively. By 4.5 it

follows then that {Dp + Tp } is an identifier satisfying the assumptions of Lemma

4.1. Similarly, by 4.6 it follows that U p + Op + T{ }, {A* + θ j } , {A{ + Γj}

are identifiers satisfying the assumption of Lemma 4.2. Finally, [Γjf } is an

identifier satisfying the assumptions of Lemma 4.2, as a consequence of 4.4. The

unessential character of all these identifiers is then a direct consequence of 4.1

and 4.2 respectively.

REMARK. The writer was unable to determine whether or not Γp coincides

5. T H E COMPLEX Γ = r(X)

5.1. Theorem 1 in 4.7 shows that any combination of the basic identification

schemes, used in previous approaches to singular homology theory, may be applied

to the singular complex R without affecting its homology structure. From the point

of view of achieving maximum reduction, the identifier {Γp } is of special interest.

We shall therefore go into some detail concerning this particular identifier. By the

general remarks made in §1, this identifier leads from the singular complex R to

a new and much smaller Mayer complex which we shall denote by r = r(X) Since

{Pp ] is unessential, r has the same homology structure as R. We want to examine

in some detail the computational facilities and conveniences available in the

complex r.

5.2. By the general remarks in §1, the elements of the p-chain group CTp of r

are of the form {cp}9 where this symbol denotes the coβet (relative to Γp) con-

taining the p-chain Cp of R. Let us adopt, in dealing with the complex r, the usual

practice of writing Cp instead of {cp}, with the understanding that Cp is now

considered as a representative of the element {cp} of CTp. For clarity, we shall

use the congruence symbol = in writing equations, to remind ourselves of the

286 TABOR RADO

fact that we are dealing actually with congruences mod Γp . We shall presently

note some of the computational rules for the complex r.

5.3. Let (v£, , v'p\ T')R, (v'όr ••, v£, T")B be two p-cells of R related

as follows. There exists a system of linearly independent points w0 , , Wp in

£ 0 0 a n d t w o l i n e a r m a p s α ' : \wo, ,wp\ — > \ V Q 9 9 Vp \9 O ί " : \w0,

• , Wp I —» I VQ , , Vp I, such that the following relations hold:

(i) α'(»i) = v'i , α"(»i) = «"i (i = 0 , , P ) ,

(ii) r α' = Γ"α" .

Then {vi , - , Vp , T')" = ( < , , Vp , T" ) R . Indeed, by the definition of A*

and Γp (see 3.4, 4.7), we have

(v'o,' ',Vp,τ')R ~ (y>o," ,vp,T'a') f

and hence

Similarly

Since Γ' α' = Γ ' ' ^ , the assertion follows.

5.4. Given a sequence v0 , , Vp of p + 1 points in £00 (which need not be

linearly independent or distinct), by a transposition we shall mean (as in §2) the

operation of exchanging two adjacent elements of the sequence v0 , , vp. Let

then (VQ, , Vp, T' ) R , (I/Q , * #» fp\ 71")* be two p-cells related as follows:

(i) \vΌ,'~,v'p\ = K •••,!/; I ,and T' = Γ "

(ii) there exists a sequence of n > 0 transpositions leading from {v'Q , ,Vp)

to (t/s , . . . , * ; ) .

T h e n K , , t ; ^ , Γ ' ) Λ Ξ ( ^ , . - - , V £ , T " ) R if n i s e v e n , and ( ^ , ,

Vp, T')R = -(V'Q, ,v'p, T")R iί n is odd. Indeed, the assertion is obvious

if 7i = 0. If n — 1, the assertion follows immediately from the fact that Tp C Γp

(see 3.2, 4.7). Repeated application of this remark yields the desired result for a

general n.

5.5. Let (vo, , Vp, T) be a p-cell such that the points vQ, , Vp are

not all distinct. Then (ι;0 , , Vp, T) = 0. Indeed, by a certain number n of

transpositions we can obtain a p-cell (u/0, >Wp» T) in which two adjacent

points WJ , Wj+X coincide. Then (see 3.3, 4.7)

(wo,--,wp, T)R C D* C f* ,

and hence

(^0, , ^ , Γ ) Λ Ξ 0 .

On the other hand, by 5.4,

(«'o, ,«'p,Γ)/1 s ±{vo, ",vp,T)Λ,

and the assertion follows.

5*6. Let (vQ , , Vp , T) be a p-cell of /?. Let w0 , , u;^, where q > p,

be a system of linearly independent points in E& > and let α : | M ; 0 , ,w;^|

—> I v0 , , Vp I be a linear map such that the points 0((M;0), , (Xdi; )

coincide with the points t>0 , , vp in any order and with any number of repe-

titions. Then

(»ormm,Vq,TOL)B Ξ 0 .

Indeed, by 5.3 we have the relation

(•o. ' . ϊ .Γα) ' 5 (a(wQ), ~,u.(wq),T)R.

On the other hand, since q > p, the points <x(wo)9 , &(wq) are not all distinct.

Hence, by 5.5, we have

and the assertion follows.

5.7. Let (v0 , , Vp, 71) be a p-cell of /?, such that the points v0 , , vp

are linearly independent. Suppose this p-cell possesses the following type of

symmetry. There exists a linear map (X : | v0 , , vp \ —> | v0 , , vp \, such

that (i) the points Ot(t;0), , (X(vp) form an odd permutation of the points v0,

• •> Vp (taken in the indicated order) and (ii) Γ(X = Γ. Then (v0, , vp, T)

= 0. Indeed by 5.4 and 5.3 we have

288 TIBOR RADO

(vo,' '',vP,T)R =-(α(υ o ),

Since T = TOL, it follows that 2(v0, , vp, T)R = 0, or equivalently

2(^o, %^,Γ)/iG ΓR.

Now since Γjf is the division-hull of Γp (see 4.7), the last relation implies the

existence of an integer k ψ 0 such that 2k (v0, , vp, Γ) G Γp , and hence

(by the definition of the division-hull) (v0 , , vp, Γ)Λ G Γp . Thus (v0 , ,

Vp, T)R = 0.

5.8. The argument just used yields obviously the general result: if ncp = 0,

where n is an integer ^ 0, then cp = 0. In other words, the p-chain group Cp of

the complex r has no elements of finite order. Of course, this is a priori obvious

from the remark that a division-hull is closed under division. It may be of interest

to determine whether or not Cp is in fact a free Abelian group. The writer was

unable to answer this question.

5.9. The homomorphisms Bp, βp9 pp , τpσp apply to congruences. In detail:

OΌ Cp = O« Cp, Pp Cp — Op Cp f pp Cp = pp Cp , ^Ό&Ό Cp = 'p&p Cp .

The first one of these asserted congruences is of course merely a restatement of

the fact that \Γp } is an identifier. The last one may be verified as follows. In

view of the identity 3.1 (4) we have

4 - 4 ) = σP 4 ~ σP CP =

and hence, by 3.4,

τpσp cp-cR

p CA*.

Since Ap C Γp , it follows that

(1) = Cp

Similarly, rpσpcR = cR. Since cR = cR, it follows that rpσpc

R = τpσpcp. Now

let us recall that μ^} satisfies the assumptions of Lemma 4.2, as we observed

in the course of the proof in 4.7. Accordingly, the assumption cp = cp , which is

equivalent to Cp — δ R £ Γjf , implies that


(2) crpβ«(c«-c«)=0,

(3) tyThe relation (3) is equivalent to PpCp = ppCp. On the other hand, (2) implies,

by 3.4, that βR

p(c$ - δ*) £A* C f/, and hence that β*c* = β*c$.

5.10. In terms of familiar terminology, the preceding results may be summarized

as follows. In the complex r, affine-equivalent p-cells of R become equal to each

other (see 5.3). The permutation rule (or the orientation convention) holds in r

(see 5.4). Degenerate p-cells of R may be discarded in r (see 5.5, 5.6), as well as

affine-symmetric p-cells (see 5.7). The operators 'dp, βp, pp continue to apply

in r (see 5.9). Furthermore, the operation τpσp is also applicable in r (see 5.9).

The effect of this operation is to replace a general p-cell (v0, , vp, T) by a

p-cell of the form (do,0 *9dp, T*) (see 0.3). Accordingly, one can avoid en-

tirely the use of p-cells (v0 , , Vp , T) where the points v0 , , vp are not

linearly independent (it is not obvious, however, that this practice, if followed

consistently, contributes to clarity and simplicity of calculations). Finally, let

us note that the complex r offers the advantage that its chain-groups do not have

elements of finite order (see 5.8). In the light of comments made in previous liter-

ature, this may represent a desirable feature.

5.11. In the course of a correspondence on these subjects, Professor S.

MacLane communicated to the writer a simple and ingenious proof of the fact

that the chain-groups of the complex r are indeed free Abelian groups (cf. 5.8).

6. CONCLUSION

6.1. One may raise the question whether the singular complex R admits of

further reductions, in terms of identifications, without affecting its homology

structure. In particular, one may ask whether there exists a maximal identification

scheme, in some natural and appropriate sense. A plausible approach may be

obtained by setting up the principle that only those identifications are admitted for

which the computational rules set forth in 5.3—5.9 hold. The problem consists then

of determining whether among all unessential identifiers {Gp}9 conforming to this

principle, there exists one, say {Gp}9 such that Gp C Gp for all identifiers {Gp}

satisfying the requirements just stated. The writer was unable to settle various

interesting questions upon which the answer to this problem seems to depend.

6.2. From a heuristic point of view, one may conjecture that, in view of the

intensive study and manifold applications of singular homology theory, it is un-

likely that any relevant identification scheme escaped the attention of the many

290 TffiOR RADO

workers in this field. For example, one may assume, as a heuristic working hy-

pothesis, that by applying simultaneously all the identification schemes used in

the papers listed in the References of the present paper one obtains a maximal

identification scheme in the sense of 6.1. The writer was unable to find a proof

for the theorem suggested by these remarks.

6.3. As regards previous literature concerned with the unessential character

of identification schemes, precise comparisons would lead to excessive detail,

particularly because our complex R has not been considered explicitly in the

literature, as far as the writer is aware. The following comments are meant to

indicate the origin of certain questions rather than the exact formulation of defi-

nitions occurring in other theories. The initial motivation for the present study,

as well as for the previous paper [6] of the writer, came from the important paper

of Eilenberg [ l] In that paper, Eilenberg shows, in effect, that (in our termi-

nology) the identifier {Tp } is unessential (see 3.2). In his previous paper [6] ,

the writer showed then that the identifier \Ap } is also unessential. However, the

unessential character of certain identifications has been recognized by various

authors. Thus Seifert-Threlfall [7] and Lefschetz [5] contain remarks suggesting

that the "affine symmetric " p-cells may be discarded without affecting the homol-

ogy structure. Tucker [δ] showed, in effect, that the system {Dp} is unessential,

at least in relation to the identifier {Tp ]. In a sense, our complex R appears thus

as the singular complex in unreduced form, alternative theories being derivable by

various types of reduction. The problems we stated in 6.1 and 6.2 amount merely

to the question whether there is some end to this process of reduction without

changing the homology structure.

REFERENCES

1. S. Eilenberg, Singular homology theory, Ann. of Math. 45 (1944), 407-447.

2. S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology (Unpublished).

3. S. Eilenberg and J. A. Zilber, Semisimplicial complexes and singular homology,Ann. of Math. 51 (1950), 499-513.

4. S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloquium Publications, vol.27; American Mathematical Society, New York, 1942.

5. , On singular chains and cycles, Bull. Amer. Math. Soc. 39 (1933), 124-129.

6. T. Rado, On identifications in singular homology theory, To appear in Rivista diMatematica della Universita di Parma.

7. H. Seifert and W. Threlfall, Lehrbuch der Topologie, B. G. Teubner, Leipzig, 1934.

8. A. W. Tucker, Degenerate cycles bound, Rec. Math. (Mat. Sbornik) 3 (1938), 287-289.

OHIO STATE UNIVERSITY

ON SOME TRIGONOMETRIC TRANSFORMS

O T T O SZASZ

l Introduction. To a given series Σ ^ = ι un we consider the transform

A sin vtnAn = 2* uv > where tn Φ 0 as n —* °°

It was shown in a previous paper [5, Section 4, Theorem 3] that the transform

(1.1) is regular if and only if

(1.2) ntn = 0 ( 1 ) , as n• o o

We shall now consider the transform (1.1) in relation to Cesaro means. In a forth-

coming paper Cornelius Lanczos has found independently that the transform (1.1)

is very useful in summing Fourier series and derived series, and gave some very

interesting examples; he takes tn — ττ/n. Of our results we quote here the follow-

ing theorem:

THEOREM 1. In order that the transform (1.1) includes {CfD summability, it

is necessary and sufficient that

(1.3) ntn=pπ + (Xn, n<Xn = θ(l), p a positive integer.

We also discuss other triangular transforms which may be generated by "trun-

cation" of well-known summation processes, such as Riemann summability. The

transform An and the transform Dn (Section 5) are special cases of the general

transform

n

Ύn = Zs

Received March 8, 1950. Presented to the American Mathematical Society December 30,1948. The preparation of this paper was sponsored (in part) by the Office of Naval Research.

Pacific J. Math. 1 (1951), 291-304.

291

292 OTTO szXsz

where φ(P) is a function of the π-dimensional point P(xι, #2> * ' *> xn)>

Pn —> 0. This transform and many special cases of it were discussed by

W. Rogosinski [4] in particular, the special case an = 0 of our Theorem 4 is

included in his result on page 96. The general approach is essentially the same

as in the present paper.

2 Proof of Theorem l If we write

" " , sinvtn sin (y + l) tn

sin vtn 2 sin {v + l) tn sin (v -f 2) t n _

(v + 1 ) ίn (v + 2 ) t n

I

then

n t

= " y s ' Δ 2 + s ' Δ + ( ' - s ' ) —

or

/0 ,x . _ n ^ 2 , Λ2 , , I sin (n - 1) tn 2 sin ntn

ntr,

sin

nntn

Now (C. 1) summability of 2 n = 1 un to 5 means that

(2.2) n - i s ^ —> s , as n —• 00 .

If sn = 1, then i4Λ = sin ίΛ/ίπ —> l

In order that (2.2) imply An —> 5, it is necessary and sufficient [in view of

(2.1)] that

ON SOME TRIGONOMETRIC TRANSFORMS 2 9 3

sin ntn , x sin (n — l) tn(2.3)

n-2

(2.4) Σ H Δ H = 0 ( 1 ) , as n

v=l

The first condition of (2.3) [in view of (1.2)] is equivalent to

sin ntn = 0(tn) = 0(l/n) ;

hence

ntn = pπ + an , ndn = θ(l) .

The second condition of (2.3) now reduces to

cos ntn sin ίn = θ ( ί n ) ,

or

cos α n sin ίn =θ(n"1) ,

which is satisfied. Finally

= / cos vx dx = K / e ι c/x

hence

(2.5) t n Δ ^ = R j Γ t n tfeivx dx=Hfo

ta eivx(l-eix)2 dx ,

and

(2.6) t B I Δ i I < JΓ t n |1 - β " | 2 dx = 4 jftn (sin x/2)2 dx

It follows that

n-2

294 OTTO szXsz

This proves Theorem 1.

We can show by an example that the transform An may be more powerful than

(C,l). In (1.3) let p = 1, nan = - π / 2 ; the series Σ * = ι (-l)n~ ι n (that is,

un — (—l)nn) is not summable (C, 1), but summable (C, 2) to 1/4. Now

in t n ~ ( - l ) n [sin ntn + sin (n + l) tn]sin

where ntn = π—7T/2n. Hence, as τι 00

An ~ 1/4 + o(l) .

An even more striking example is un

= ("~l)n n2 .

3. Summation by harmonic polynomials. We get a more powerful method if we

introduce the harmonic polynomial

and the corresponding transform

(3.2) Bn= Σ »vPn

or

βn = tn

ϊhn(pnf tn)

Let

n

Sn = Σ

where

fc _ ( k ' + l ) ••• {k + n ) nk

Ύn =n! Γ(k + 1) '

we also write

and

σ*-fLn y

ΎnNow {C,k) summability of the sequence [sn] to s is defined by

lim σ\f = s •

We quote the following elementary theorem [cf. 6, Theorem l ] , which is included

in a more general result of Mazur [ l , Theorem X] :

LEMMA 1. Let k be a given positive integer^ and let

n = 0,1,2, •••.

In order that lim Tn exist, whenever the sequence {sn} is \C9k)summable to s,

it is necessary and sufficient that:

n

<3 3 ) Σ Ύv \&antv I = 0(1) , α M = 0 f o r ^ > n ;

^ ^ lim y ί Δ α π > v = α v βΛ ί s ί s , v — 0, 1, 2,n-»oo

S) lim V απ v = /3 exists.π-»α> ^^ '

We then have lim Tn — βs + Σ^=o CXv(σv ~"-s) Since then the transform ΓΛ

296 OTTO SZASZ

is convergence preserving we must have (3.5) and:

lim anv exists,n-»co

V = 0 Ί 2" υ > J-f *>t t

hence (3.4) and (3.5) hold, so that the conditions of Lemma 1 reduce to (3.3). In

the case of the transform Bn, we have

«π,n = Pitsin nt n

sin {y + l) tn

(* + l ) t Λ '

hence

To satisfy (3.3) we must have

(3.6)

(3.7)

0 ,

sin nt

n pn

-i sin (n -\) tn

77{n - 1) tn

= 1,2,

as

and

(3.8)

k n-k s i n (n

P ~7(π -fe) tn

n-k-ls in

= 0(1)

_ sin ntn

Assume first that k = 0 then our conditions become:

(3.9)

and

(3.10) - -" S i n Vtn - S i Π {V + 1 } t n = 0(1)

ON SOME TRIGONOMETRIC TRANSFORMS 297

We now prove the lemma:

LEMMA 2. / /

(3.11) pS- ••— ρn

α s t n Φ 0 ,

then Rn is a regular transform.

Clearly (3.9) holds, and we need only to show that (3.10) also holds.

If pn > 1, then p% < p%, v - 0, 1, , n - 1 if on the other hand pn < 1,

then p% < 1. Hence, in either case,

max pi = 0(1) ,0<v<n as n

00

We have

sin vt sin (v + l) t~ P

v v + 1<

sin vt sin (v + l ) t

Σs i n

+ 1

+ l ) t

v + 1

the second term is O(t), and

sin (v + l) ts m

-f 1= Γ° cos (v = 0{t2) ,

so that

ΣPV sin vt sm

+ 1= 0

Thus (3.10) is satisfied and Lemma 2 holds.

Note that the condition p% = 0(1) is equivalent to n(pn ~ 1) < c, a positive

constant (see [5>p. 73]); furthermore, if ntn = 0(1), then clearly the secondcon-

dition of (3.11) holds.

Next let k — 1 we shall prove the theorem:

298 OTTO SZASZ

THEOREM 2. //(3.11) holds, and if

(3-12) PX s i n ntn=O(tn),

then Bn includes (C91).

The conditions (3.6)—(3.8) now become :

p5 sin ntn =O(tn) ,

pZ sin (n - 1) tn =O(tn) ,

CO

and

(3.13)n-2

Σv-l

sin vu

v= o{tn), as n • • o o

Clearly, we need only to show that (3.13) is satisfied. Now

sin vtΔ2 pι = Δ 2 / f* cos vx dx = RΔ2 f* pveivx dx

= Hfo

tpveivx(l-2peix +p2e2ix) dx

= Kft pveivx(l-peix)2 dx

Hence

sin vt

v<PV Γ \ 1 - P e ί x \ 2 d x ] Σ

This proves (3.13) and Theorem 2.

4. Comparison of Bn and (C, k), k > 2. We wish to prove the following theo-

rem :

ON SOME TRIGONOMETRIC TRANSFORMS 299

THEOREM 3. Suppose that (3.11) holds and that

(4 D n*"VS sin n t B = O ( t B ) ,

(4.2) nk~ιρ$ cos ntn =0(1) ,

then Bn includes (C,k) summabilίty.

pnt Λ Φ 0 ,

Now (3.6) holds because of (4.1), and then (3.7) follows from (4.2). It remains

to prove (3.8). We have

hence

(4.3)

sin

V J*

•pvsin vt

<pvfo

t | i -/

-peίx)k+1 dx;

It follows that

(4.4)v = l

sin vtn

Pnvtn

Λ+i]

= 0 (l-Pn)k+1 Σ ^ +0 K

Here the first term is 0(1) by Lemma 2 of [ό] finally

-0(1)

300 OTTO SZASZ

This proves Theorem 3.

An interesting special case is tn = 7τ/n; the conditions now reduce to the

single condition

If, in particular, nkp% — 0(1) for all k9 then Bn includes all (C, k).

Observe that by Lemma 1 of [6] the condition n p% — 0(1) is equivalent to

lim sup \n(pn - l ) + fe log n] < +«> .

Note also that (4.1) and (4.2) imply:

n*-VS = 0(1) .

5 Truncated Riemann summability The series Σ v=0 uv is called (R9k)

summable to s if the series

(5.1)CO / . Λk

^ / s i n nt\ , N

+ Σ I — I "IE =Rk(t)n = l nt

converges in some interval 0 < t < t0, and if

Rk(t)—>s, as t •0.

For A; = 1 it is sometimes called Lebesgue summability. The method (/?, k) is

regular for k > 2 and, in fact, it is more powerful than (C9 k — 2) for k = 2, it

was employed by Riemann in the theory of trigonometric series. We generate from

it by truncation the triangular series to sequence transform {u0 — 0):

sin vtn

n - l

= Σsin vtn sin ntr.

ntr

k is a positive integer. We assume k > 2; it is then easy to show that Dn is a

regular transformation.

From Lemma 1 we find for (C9 k) to be included in Dn the conditions:

(5.2)

(5.3)

t;* (sin ίΓ^TtJ* =0(1), for v = 0,1, * , k

n-k-lsin vtn = 0(1), * oo

It follows from (5.2) (see Section 2) that we must have

(5.4) ntn - pπ + 0Ln , n an = θ(l) , p a positive integer

now (5.2) reduces to

tn sin (θLn~vtn) = θ(l) , V = 0, 1, , k ,

and this is satisfied in view of (5.4).

To show that now (5.3) also holds, we employ a lemma, due to Obreschkoff

[2,p. 443]:

LEMMA 3. We have

sin vt

vt<M

v

where M is independent of t and V.

It now follows that

Σ ^sin vtn) = O(ntn) =0(1), > 0 0

This yields the following theorem:

THEOREM 4. If ntn — pu + ctn, p a positive integer^ n<Xn — 0(1), then the

transform

JL lsinvtn\k _.Λ

includes {C9k) summability (k a positive integer).

6. A converse theorem* We shall establish the following result.

THEOREM 5. //

k(6.1) lim inf

sin ntn = λ > 1/2 ,

then the transform Dn is equivalent to qonvergence.

302 OTTO szXsz

It follows from (6.1) that lim sup ntn < 2i/k hence (see Sections 1 and 5) the

transform Dn is regular. We now wish to show that Dn —> 5 implies sn —» s;

we follow a device used by R. Rado [3] •

Assume first that s = 0, and that sn — 0(1); then

0 < lim sup \sn I = δ < °o fπ-»oo

and we shall show that S = 0. To a given e > 0 choose n — n(β) so that js v | <

8 + € for v > n. Next choose m > n and such that \sm\ > δ — £. We have

sin mt.

where

Jfίtn

sin vtnin iy + 1) Vfsin

+ 1) t

hence, as mt < 77, we have

s in

/nt-

m - 1

Σ

<o(l) + (δ -h e) fsm ntnf s in mtn

nit.

It follows that

δ - β < Is, I < o ( l ) + (δ + e ) {1/λ- 1 + o ( l ) } .

But l /λ < 2, and € is arbitrarily small; hence δ = 0.

We next assume s = 0 and lim sup \sn\ — °° choose € > 0 and ω large.

Denote by m = m(α ) the least m for which | sm \ > ω; then

ω< \sm\ < o ( l ) +

But this is impossible for λ > l/2, small e, and large m.This proves our theorem

for s = 0. Finally, applying this result to the sequence \sn ~~ s |and its transform

completes the proof of Theorem 5.

7. Application to Fourier series* Suppose that f(x) is a Lebesgue integrable


function of period 277, and let

GO

(7.1) f(x) ~ αo/2 + Σ (αn c o s n x "*" bn s i n nx) = Σ u n (*) ί

we may assume here α0 = 0. Now (cf. [7,p. 27])

00 ^

F(X) — f f(t) dt = C + ^ (aa sin nx ~~ 6 a cos ΠΛ) — ,0 γi

where

00 -,

c = ? „ n

It is known [7, p. 55] that at every point x where F'(x) exists and is finite, the

series (6.1) is summable (C?r), r > 1, to the value F'(x).

It now follows from Theorem 3 for k — 2 and tn — τr/n that if np\ ~ 0(1), then

" v sin vπ/n t

Furthermore, Theorem 4 yields, for k — 2, that if

ntn = prr + α n , nα n = θ(l) ,

then

n /sin vίn\2

uv \x) i i ' r V /

An analogous theorem holds for higher derivatives (cf. [7, p. 257] ).

304 OTTO SZA'SZ

REFERENCES

1. St. Mazur, Uber lineare Limitierungsverfahren, Math. Z. 28 (1928), 599-611.

2. N. Obreschkoff, Uber das Riemannsche Summierungsverfahren, Math. Z. 48 (1942-43), 441-454.

3. R. Rado', Some elementary Tauberian theorems (I), Quart. J. Math., Oxford Ser. 9(1938), 274-282.

4. W. Rogosinski, Abschnittsverhalten bei trigonometrischen und Fourierschen Reihen,Math. Z. 41 (1936), 75-136.

5. Otto Szasz, Some new summability methods with applications, Ann. of Math. 43(1942), 69-83.

6. , On some summability methods with triangular matrix, Ann. of Math. 46(1945), 567-577.

7. A. Zygmund, Trigonometrical series, Monografje Matematyczne, Warszawa-Lwow,1935.

NATIONAL BUREAU OF STANDARDS, LOS ANGELES

ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS

J. G. WENDEL

l Introduction. Let G be a locally compact group with right invariant Haar

measure m [29 Chapter XI]. The class L(G) of integrable functions on G forms a

Banach algebra, with norm and product defined respectively by

IWI=/U(g) !

The algebra is called real or complex according as the functions x(g) and the

scalar multipliers take real or complex values.

Suppose that T is an isomorphism (algebraic and homeomorphic) of the group G

onto a second locally compact group Γ having right invariant Haar measure μ;

let c be the constant value of the ratio m(E)/μ(τE), and let χ be a continuous

character on G. If T is the mapping of L (G) onto L (Γ) defined by

(Tx)(τg)=cχ(g) χ(g), xCL(G),

then it is easily verified that Γ is a linear map preserving products and norms;

for short, T is an isometric isomorphism of L (G) onto L (Γ).

It is the purpose of the present note to show that, conversely, any isometric

isomorphism of L (G) onto L(Γ) has the above form, in both the real and complex

cases.

We mention in passing that if T is merely required to be a topological iso-

morphism then G and Γ need not even be algebraically isomorphic. In fact, let G

and F be any two finite abelian groups each having n elements, of which k are of

order 2. Then the complex group algebras of G and Γ are topologically isomorphic

to the direct sum of n complex fields, and the real algebras are topologically iso-

morphic to the direct sum of k + 1 real fields and (n — k — l)/2 two-dimensional

algebras equivalent to the complex field. The algebraic content of this statement

Received October 24, 1950.

Pacific /. Math. 1 (1951), 305-311.

305

306 J. G. WENDEL

follows from a theorem of Perlis and Walker [4] , but for the sake of completeness

we sketch a direct proof.

Since the character group of G is isomorphic to G there are exactly k characters

Xi 9 X2 > # # " f Xk on G of order 2. Together with the identity character χ 0 these

are all of the characters on G which take only real values. The remaining charac-

ters Xk + ι> * ' •» Xrc-i f a l l i n t 0 complex-conjugate pairs, χ 2 m = χ 2 m + i> ™ =

(k + l)/2, (k + 3)/2, • • • , ( « - 2)/2. For 0 < y < n - 1 let Xj £ L (G) (complex)

be the vector with components (l/rc)χy(g). It is readily verified that the Xj are

orthogonal idempotents, so that L (G) can be written as the sum of n complex

fields, and the same holds for the complex algebra L ( Γ ) . In the real case we

retain the vectors XJ for 0 < / < k, and replace the remaining ones by the (real)

vectors ym — x2m + *2m + t 9 zm ~ iχim ~~ iχ2m + ι » whose law of multiplication is

easily seen to be yl = ym , z2

m = - y m , ymzm = zmym = z m , while all other

products vanish. Since the vectors xj, y m , 2rm span L (G) we see that L (G) is

represented as the sum of k + 1 real fields and (n — & — l)/2 complex fields; the

same representation is obtained for the real algebra L(Γ) ; this completes the

proof of the algebraic part of the assertion. The fact that these algebras are also

homeomorphic follows from the fact that all norms in a finite dimensional Banach

space are equivalent.

2. Statement of results* For any fixed g0 £ G let us denote the translation

operator x(g) —> x(golg)i x C L{G), by SgQ; operators Σγ are defined

similarly for L ( Γ ) . In this notation our precise result is:

THEOREM 1. Let T be an isometric isomorphism of the (real, complex) algebra

L {G) onto the (real, complex) algebra L (Γ). There is an isomorphism r of G onto

F, and a {real, complex) continuous character X on G such that

(1A) TSST'1 = χ ( g ) Σ τ g , g G G,

(IB)* (Γ*)(τg) = c χ(g) x(g), g G G, x CL(G) ,

where c is the constant value of the ratio m{Ej/μ(τE)

For the proof we make use of a theorem due to Kawada [3] concerning positive

*I am obliged to Professor C. E. Rickart for suggesting the probable existence of aformula of this kind.

ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS 3 0 7

isomorphisms of L (G) onto L(Γ) in the real case; a mapping P : L (G) —» L(Γ)

is called positive in case x(g) > 0 a.e. in G if and only if (Px)(γ) > 0 a.e. in

Γ. Kawada's result reads:

THEOREM K. Let P be a positive isomorphism of L (G) onto L (Γ), both alge-

bras real. There is an isomorphism r of G onto Γ such that PSgP~ι = kgΣTg,

g G G9 where kg is positive for each g.

In order to deduce Theorem 1 from Theorem K we need two intermediate results,

of which the first is a sharpening of Kawada's theorem, while the second reveals

the close connection which holds between isometric and positive isomorphisms.

THEOREM 2. Let P be a positive isomorphism of real L(G) onto L(T)*Then:

(2A) P is an isometry;

(2B) kg = 1 for all g C G;

(2C) P is given by the formula (Px){rg) = cx(g), where c is the constant value

of the ratio m(E)/μ(τ E)

THEOREM 3. Let T be an isometric isomorphism of L(G) onto L(V). There is

a continuous character χ(y) on Γ such that if the mapping P : L (G) —ϊ L (Γ) is

defined by {Px)(y) = χ(y){Tx)(γ), x € L(G), y C Γ, then P is a positive

isomorphism of the real subalgebra of L (G) onto the real subalgebra of L (Γ). The

character X is real or complex with L (G) and L (Γ).

3 Proof of Theorem 2. P and its inverse are both order-preserving operators,

and therefore are bounded [ l , p 249] Consequently the ratio \\Px \\ /\\x || is

bounded away from zero and infinity as x varies over L (G), x ψ- 0. If x is a posi-

tive element of L (G) it follows by repeated application of Fubini's theorem that

\\xn\\ - \x\n\ since Px is also positive, and P (xn) = (Px)n, we have the result

that for fixed positive x ^ 0 the quantity {\\Px \\/\\x \\}n is bounded above and

below for n — 1,2, . Hence P is isometric at least for the positive elements

of L (G). But now for any x C L(G) we may write x = x + x , where x and x

denote respectively the positive and negative parts of x. Then

11*11 = llχ+ + χΊI = ll«+ll +ιl*ΊI = llΛc+|| + ||Λtl>||ft+ + p«Ί| = ||fte||.

Applying the argument to P~ι we obtain the result

11*11 " I I

308 J. G. WENDEL

which is the statement (2A).

Theorem (2B) follows at once from this and Theorem K. For if x £ L (G) then

IISgX II = TOg||#||, where πig is the constant value of the ratio m(gE)/m(E). Simi-

larly, | | Σ T g ^ : | | = μTg \\ζ\\ Since r is a homeomorphism,μ τg — mg.The constant

kg may now be evaluated by taking norms on both sides of the equation PSgP~ι

= kgΣTg, and must therefore have the value unity.

To prove part (2C) of the theorem we observe that the operator Q defined by

(Qx)(τg) — cx(g) satisfies the relation QSgQ~x — Σ T g, and is an isomorphism of

L (G) onto L (Γ) . Then QSgQ~ι = PSgP~ι, g £ G, and consequently R = P~ιQ

is a continuous automorphism of L (G) which commutes with every Sg. We shall

show that R must be the identity mapping.

Segal [5, p. 84] has shown that the product xy of two elements x9 y belonging

to L (G) may be written as a Bochner integral, which in our notation takes the form

xy= Jx(h)mlι{Shy}m(dh),

where the quantity in braces is a vector-valued function of h £ G9 and the function

mg was defined above. Applying the operator R we obtain

R(xy) = $x{h)*iι{RShy\m(dh) = SxWm^iS^ylmidh) =xRy.

But R is an automorphism, and so also R (xy) — (Rx)(Ry). Thus x = Rx, all

x G L (G), which shows that P — Q9 as was to be proved.

4 Proof of Theorem 3 We first require several lemmas, all of which share the

hypothesis: T is an isometric isomorphism of L (G) onto L ( Γ ) , indifferently real

or complex. For x, γ £ L (G) we write ξ for Tx, 7) for Ty. We denote by E (x) thes e t £g|& €1 G9x(g) 7^ θ}9 which is regarded as being determined only up to a

null-set; E (ζ) in Γ is defined in the same fashion. (Although we make no use of

this fact, the first three lemmas below actually hold in case T is an isometry

between two arbitrary L-spaces.)

LEMMA 1. If EM Π E (y) = Λ then E {ξ) Π E (17) = Λ , and conversely.

Proof. The hypotheses imply that for all scalars A we have \x + Ay\\ — \\x\

+ \A\ \\y\\. Then for all A we have \\ξ + AΎ)\\ = \\ξ\\ + \A\ | | η | | , which implies

that E {ξ) and E (η) are disjoint. For the converse we need only replace T by T *.

LEMMA 2. IfE(x) C £ (y) then E(ξ) C £ (η), and conversely.

Proof. Suppose that E (x) C E (y), but that E (ξ) $ E(η). Then we may

write ξ= ξx + ξ29 w i t h f i ^ ) C £(η), £ ( £ 2 ) Π £ ( η ) = Λ = E (ξ x) Π E (ξ2).

Let Γ x ^ = xι\ then from Lemma 1 it follows that E(xχ) Π E(x2) — A = £Gt2)

Π £(y). But E{xχ) U E(x2) = E (x) C £ (y); this contradiction yields the result.

LEMMA 3. Let B in Γ be a σ-finite measurable set {that is, the sum of a

countable number of sets of finite measure). Then there is a positive x C L (G)

such that E {ξ) - B.

Proof. Let 77 £ L{Γ) be chosen so that £ (η) = B. Let y = T~ιη, and setΛ Q>) = ITQ>) I > £ £ G. Then Λ C L (G)> E{X) — E (y), and therefore from Lemma

2 it follows that £ (£) = β.

LEMMA 4. Le£ A; o/icί y be positive elements of L(G). For y C E{ξ) let

Kξ (γ) = ξ(y)/\ζ{y)\9 and define Kv (y) in similar fashion. Then Kξ (y) =

Kv (γ) almost everywhere on E (ξ) Γ) E (η).

Proof. Since x and y were taken to be positive we have \\x + y | |= ||Λ;|| + ||y||.

Therefore \\ξ + η | | = | ^ | | + |[^H. Then \ξ(γ) + η(Ύ)\ = \ξ(γ)\ + \V(γ)\

a.e. in F . Hence, since the functions K have modulus 1,

\κξ(y)Kη(γΓ1\ξ(y)\ + \V(Ύ)\\ = +a.e. in £ (ξ) Π E (η). But then Kξ {y)Kv (y ) " 1 = 1 a.e. on £ (£) Π £ (η), as was

to be proved.

LEMMA 5. There is a unique continuous character χ on Γ with the property

that for all positive x G L(G) we have ξ{y) = X ( y ) | £ ( y ) | a.e.\ χ is reaZ or

complex with L (G) and L (Γ).

Proof. Let Fo be the open-closed invariant subgroup of Γ generated by a

compact neighborhood of the identity. Since Γo is σ-finite we may apply Lemma 3

to obtain a positive x £ L{G) such that £ (ξ) — Γo . Now x > 0 implies that

| |* 2 I = I* I 2 ; then also ||<f 2 ' | | = ||<f | | 2 . The element ξ2 is given by the formula

Since * 2 is also positive we have from Lemma 4 that X^ 2 (y) = ^ ^ (y) a e o n

E(ξ2) Γ\ E(ξ) C Fo = E{ξ). Writing simply K{y) for the common value, we see

310 J. G. WENDEL

that the relation ξ2{y) = K(γ)\ξ2(γ)\ therefore holds in Γo even outside ofE (ξ2). Then

\ξ2{y)\ = JTo

Integrating over Γo again we obtain

= U\\2 = hidy) J

Therefore K(γ)~ιK(γδ~ι)K(8) = 1 a.e. on Γo X Γo. Then there is a null-set

i V C Γ 0 such that y €[ N implies K (yS"1) K(8) = K(γ) for almost all 8 G Γo.

We integrate this equation over a set M of finite positive measure and obtain

K(Ύ)μ(M) = JVo

φu(Sγ)μ(d8),

where φy is the characteristic function of M. The right member is easily seen to

be a continuous function of 7, for all 7 £ Γo hence £ ( 7 ) is equal a e. to a

continuous function Xo(7)> which is clearly a character on Γo. From Lemma 4 it

follows also that, for positive x £ L (G), if E (ξ) 9 T o then ξ(γ) = χ 0 (7)

The proof is completed by extending the function χ 0 to all of Γ. To do this we

write Γ as the union of disjoint cosets 7αΠ)> a n ^ consider the open-closed sub-

group Γ\ generated by any finite number of cosets. Then Γ\ is again σ-finite, and

we may repeat the above argument to obtain a continuous character )(i on Fj

Lemma 4 guarantees that for two such subgroups Γ\ and Γλ the characters )( t and

Xi will agree on Γx Π Γ/ 2 Π)> s o ^at Xi ^s indeed an extension of χ 0 . Clearly,

i f * > 0 a n d £ ( £ ) C Γt then ^(7) = χ ^ γ ) | ^(7) | .

Finally, X on all of Γ is defined by χ(y) = χ x (y) for y £ Γi Since the union

of all such subgroups ΓΊ is precisely Γ, and since as shown above the subgroup

characters are mutually consistent, the function X is well-defined. It is clearly a

continuous character. The remaining property, that x > 0 implies ζ{y) = χ ( y )

l ί ί y ) ' * can be proved as follows. The set E (ζ) intersects at most a countable

number of cosets ynΠ> ^n s e t s °f positive measure. Let ξn be the restriction

to ynV0 of ξ, and put xn = T~ιξn. Then x = Σn=ιXn, and by Lemma 1 the sets

E{xn) are pairwise disjoint, so that the xn are themselves positive elements. From

this it follows that^(y) = χ Λ ( y > | £ n ( y > | = χ(y)\ξn(y)\ f o r r G r Λ r 0 ;

hence the result holds.

The proof of Theorem 3 is now immediate. For the continuous character X on

Γ constructed in Lemma 5 the mapping P on L (G) to L (Γ) defined by (JPx){γ)

= χ(y)~ι (Tx) (y) carries positive elements of L (G) into positive elements of

L (Γ); P is clearly an algebraic isomorphism of L (G) onto L (Γ). We have only to

show that Px positive implies x positive. Suppose then that Px = ξ is positive,

but that x = xx ~ x2 + i (x3 - * 4 ) , with XJ > 0 and E (xx) Π £ 0c2) = £ (x3) Π £ ( * 4 )

= Λ, and correspondingly ξ = ξγ ~ ^ 2 " ι (^3 "~ ^4)* ^s evidently an isometry,

and therefore by Lemma 1 the sets E(ξι) Π £(^2) a n ( l E(<ξ3) Π E(ξΛ) are null-sets.

Therefore ^ 2 = ^ 3 = ^ 4 = 0 ; so x ~ Xι 9 and x is positive.

5 Proof of Theorem 1. Because of Theorem 3 we may apply Theorems Kand

(2B) to the real sub-algebras of L(G), L (Γ), to conclude that there is an iso-

morphism T of G onto Γ such that PSgP~ι — Σ T g . Since r is a homeomorphism we

may regard the function X as a continuous character on G, by defining χ(g) =

χ(rg). By Theorem (2C), P is given on the real subalgebras by the formula (Px)

(rg) = ex (g), and, because of the linearity, this formula must hold throughout all

of L (G). Therefore (Tx)(rg) = cχ(g) x{g), which proves (IB). Theorem (1A) is an

easy consequence of this formula.

We note finally that Theorem (2A) shows that Kawada's theorem follows from

Theorem 1.

REFERENCES

1. Garrett Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publications, vol.25;American Mathematical Society, New York, 1948.

2. P. R. Halmos, Measure Theory, D. Van Nostrand, New York, 1949.

3. Y. Kawada, On the group ring of a topologicaί group, Math. Japonicae 1 (1948), 1-5.

4. S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer.Math. Soc. 68 (1950), 420-426.

5. I. E. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Soc.53 (1947), 73-88.

YALE UNIVERSITY

ON THE Lp THEORY OF HANKEL TRANSFORMS

G. M. W I N G

l Introduction. Under suitable restrictions on f(x) and vy the Hankel trans-

form g(t) of f(x) is defined by the relation

ω g ( t ) = $* {χ

The inverse is then given formally by

(2) / ( * ) = J f (xtY/2Jv(xt)g(t)dt.

These integrals represent generalizations of the Fourier sine and cosine trans-

forms to which they reduce when V = i: 1/2. The L p theory for the Fourier case

has been studied in considerable detail. In this note we present some results con-

cerning the inversion formula (2) in the Lp

a case.

It is clear that if f(x) £ L and H(v) > —1/2 then the integral in (1) exists.

It has been shown [3,6] that if f{x) ζlLp,l<p<2, then

converges strongly to a function g (t) in L p . For this case Kober has obtained the

inversion formula,

/(«) == ..-1/2-V A.

dxv+1/2 ί

«, (xt)1/2 Jv+t(xt)

which holds for almost all x In her investigation of Watson transforms, Busbridge

[ l ] has given analogous results for more general kernels. Except when p = 2

the question of the strong convergence of the inversion integral has apparently

been considered only in the Fourier case [2] . We now investigate this problem

Received January 10, 1951.

Pacific J. Math. 1 (1951), 313-319.

313

3 1 4 G. M. WING

for the Hankel transforms. We assume throughout that R(v) > —1/2.

2. Theorem. We shall establish the following result.

THEOREM 1. Let fix) G Lp

9 1 jv{χt)g{t)dt,

then

fa(x) G Lp and /(*) = Li.m. fa (x) .

Proo/. Write

/.(* .* )= / o

α (χt)1/2

(xu)1/2f(u)dufo

aJv(ut)jv(xt)tdt.

Since gbit) converges in the mean to git) it follows that lim^o, faix$b) = faiχ)

Hence

where [9]

(5) ίT(x,ufα)= Jfα Jv{ut)Jv(xt)tdt

An integral very similar to (4) has been studied in a previous paper [lO] . The

same methods may be used here to show that || /a(*)||p < Λίp|| /(#)||p Our theorem

will now follow in the usual way if we can prove it for step functions which vanish

outside a finite interval. Let φix) be a step function, φix) = 0 for x > A, and

let φaix) correspond to it as in (4). Choose ξ > 2A9 a > A, to get

Jg* \Φa(x) -Φ{x)\pdx = •// dx )// φ(u)(χUy* K(x,u,a) da \".

ON THE Lp THEORY OF HANKEL TRANSFORMS

From the relations

(6) xin Jv{x) = (2/π)1/2 {cos (* + K) + x~lAv sin (* + K)\ + θ{x~2)

(«-•»)..

where

Av = (l - 4v2)/8 , K =-(2v+ l )τr/4,

and

(7) Jv{x) = O(xvi) (* —» 0 ) ,

where 1^ = R (v), it is easy to see that

so that we have

/ / \Φa(x) -φ

for ξ sufficiently large. Now

As α —» 00 the integral goes to zero by the L2 theory for Hankel transforms

(see [7, Chapter 8J ). This completes the proof.

3 The case p — 1. Theorem 1 fails to hold in the case p = 1. The proof,

similar to that given by Hille and Tamarkin in the Fourier case [2] , will only be

sketched*

THEOREM 2. There exists a function h(t), the Hankel transform of a function

φ{x) £ Lysuch that if

(8) Ψa(x)= fo

a ( * 0 1 / 2 Ju(xt)h(t) dt

then l.i.m. ψa(x) fails to exist.

3 1 6 G. M. WING

Proof. Let h(t) = tί/2 /v(ί)/log(ί + 2). Two integrations of (8) by parts and

use of formulas (5), (6), and (7) yield

( 9 ) ^ α W ( 2 Λ\ Ί

U^-l) logfor large Λ; .

Now define ι//(#) = limα-co ψaix). It is evident from (8) that i/>(x) is con-

tinuous except perhaps at Λ; = 1, while (9) shows that ψix) = 0 ( # " 2 ) . To show

that ι//(#) C L it suffices to consider the neighborhood of x = 1. Formula (6)

yields, after some calculation,

/ \ /•<» cos (1 — x) t , xΨ W = ί log (, + 2) " ' + α ( ' ' '

where <χix) is continuous near x = 1. Thus

dt+ fW 5 J° ί log (2 + t/β) J o t log (2 + t)

The first integral on the right tends to zero as € —» 0 . Since ψix) ~" CC(%) is

positive (see [2] ) it follows that ψix) — OC(Λ ) is integrable over (1,2) [β,

p. 342] . The interval (0,1) may be handled similarly. Hence ψ(x) C L .

That hit) is indeed the Hankel transform of ψix) is a consequence of a result

of P. M. Owen [5,p.31θ] . But it may be seen from (9) that ψaix) is not in L , so

that l.i.m ψaix) surely fails to exist.

4. A summability method. It is natural to try to include the case p = 1 into

the theory by introducing a suitable summability method. Our interest will be con-

fined to the Cesaro method. If fix) £ L and git) is its Hankel transform then we

shall define

/.(*) = fo° (1 - t/a)k{xt)U* Jv(*t)g(t) dt

= C f(y)ck(χ,y,a) dy,

ON THE Lp THEORY OF HANKEL TRANSFORMS 317

where

(11) Ck(x,y, a) = f* (xy)1/2 uJ v(*α)j v(yu)(l - u/a)k da.

Offord [4] has studied the local convergence properties of fa(x) for k = l.We

are able to extend his results to the case k > 0, but the estimates required are

too long and tedious for presentation here. Instead we investigate the strong con-

vergence.

THEOREM 3. Let fix) G L, k > 0. // faix) is defined by (10), then faix)

converges strongly to fix).

Proof. We shall first prove that Cjcix9y9a) C L and ||C&(#>y,α)|| < M,

where the norm is taken with respect to x and the bound M is independent of γ and

a. An integration by parts and a change of variable in (11) give

/Ί o) C ( \ — I (Λ — λκ~"l ( \l/2 Π J

2 ^

where

= Jy+i(ays)Jv(axs) - Jv

y - x

Jv+ι{ays)jy{axs) + Jy{ays)jv+ι{axs)*~ .

Consider

where

Jv+1(ays) (θ < s < l),

(s > 1).

318 G. M. WING

Now, as a function of s , G(a9y9s) £ Lp for some p > 1 so that

F(a,y,z) = fΰ

ωG(a,y,s)(sz)1/2Jv{sz)ds

is in Lp as a function of z [3] Also

a'<Ap JjΓ" \G(a,y,s)\Pdsy/P <M,

where M is a constant independent of a and y. Thus

The other parts of (12) may be cared for similarly, so that we have

The range | y — x \ < I/a is easily handled since, by (11), for this range we have

y,α) | < Ma- Hence | |C^(x,y,a)| | < M. We see at once from (10) that

C \fa(χ)\dx= tfdx

< C \f(y)\dy / / \Ck(x,y,a)\dx,

so ||/a(^)|| ^ ^l l/(*) | | The proof may now be completed by the methods of

Theorem l

REFERENCES

l I. W Busbridge, A theory of general transforms for functions of the class Lp(0,«>)(1 < p < 2), Quart. J. Math., Oxford Ser. 9 (1938), 148-160.

2. E. Hille and J. D. Tamarkin, On the theory of Fourier transforms, Bull. Amer. Math.Soc. 39 (1933), 768-774.

3. H. Kober, Hankelsche Trans format ionen, Quart. J. Math., Oxford Ser. 8 (1937),186-199.

4. A. C. Offord,0τι Hankel transforms, Proc . London Math. Soc. (2) 39 (1935), 49-67.

ON THE Lp THEORY OF HANKEL TRANSFORMS 319

5. P. M. Owen, The Riemannian theory of Hankel transforms, Proc. London Math. Soc.(2) 39 (1935), 295-320.

6. E. C. Titchmarsh, A note on Hankel transforms, J. London Math* Soc. 1 (1926),195-196.

7. , Introduction to the theory of Fourier integrals, University Press, Oxford,1937.

8. r The theory of functions, University Press, Oxford, 1932.

9. G. N. Watson, Theory of Bessel functions, University Press, Cambridge, England,1922.

10. G. M. Wing, The mean convergence of orthogonal series, Amer. J. Math. 72 (1950),792-807.


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Pacific Journal of MathematicsVol. 1, No. 2 December, 1951

Tom M. (Mike) Apostol, On the Lerch zeta function . . . . . . . . . . . . . . . . . . . . . . 161Ross A. Beaumont and Herbert S. Zuckerman, A characterization of the

subgroups of the additive rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Richard Bellman and Theodore Edward Harris, Recurrence times for the

Ehrenfest model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Stephen P.L. Diliberto and Ernst Gabor Straus, On the approximation of a

function of several variables by the sum of functions of fewervariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Isidore Isaac Hirschman, Jr. and D. V. Widder, Convolution transforms withcomplex kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Irving Kaplansky, A theorem on rings of operators . . . . . . . . . . . . . . . . . . . . . . . 227W. Karush, An iterative method for finding characteristic vectors of a

symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Henry B. Mann, On the number of integers in the sum of two sets of positive

integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249William H. Mills, A theorem on the representation theory of Jordan

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Tibor Radó, An approach to singular homology theory . . . . . . . . . . . . . . . . . . . . 265Otto Szász, On some trigonometric transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 291James G. Wendel, On isometric isomorphism of group algebras . . . . . . . . . . . 305George Milton Wing, On the L p theory of Hankel transforms . . . . . . . . . . . . . 313

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