Proof Riemann

8/10/2019 Proof Riemann

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A PROOF OF THE RIEMANN HYPOTHESIS

Louis de Branges*

Abstract. A proof of the Riemann hypothesis is obtained for the classical Euler zeta func-tion, for its generalization with Dirichlet characters, and for Hecke zeta functions constructedin Fourier analysis on skewfields. The inverse problem for the vibrating string is treated inpreparation for Fourier analysis on the complex plane and its quaternionic generalization, thecomplex skewplane. The proof of the Riemann hypothesis for Hecke zeta functions combines

Fourier analysis for the complex skewplane with Fourier analysis for padic skewplanes. Aproof of the Riemann hypothesis is obtained follows for Euler and Dirichlet zeta functions.

1. The Inverse Problem for the Vibrating String

The Riemann hypothesis is a conjecture concerning the zeros of a special entire function,not a polynomial, which presumes a relationship to zeros found in polynomials. TheHermite class of entire functions is a class of entire functions which contains the polynomialshaving a given zerofree halfplane and which maintains the relationship to zeros found inthese polynomials.

A nontrivial entire function is said to be of Hermite class if it can be approximatedby polynomials whose zeros are restricted to a given halfplane. For applications to thevibrating string the upper halfplane is chosen as the halfplane free of zeros. If an entirefunction E(z) ofz is of Hermite class, then the modulus ofE(x+iy) is a nondecreasingfunction of positive y which satisfies the inequality

|E(x iy)| |E(x + iy)|

for every real numberx. These necessary conditions are also sufficient. The Hermite classis also known as the Polya class.

An entire function of Hermite class which has no zero is the exponential

exp F(z)

of an entire function F(z) ofz with derivative F(z) such that the real part of

iF(z)

*Research supported by the National Science Foundation

1


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2 LOUIS DE BRANGES December 18, 2013

is nonnegative in the upper halfplane. The function

iF(z) = a ibz

is a polynomial of degree less than two by the Poisson representation of functions whichare nonnegative and harmonic in the upper halfplane. The constant coefficient a hasnonnegative real part and bis nonnegative.

If an entire function E(z) of z is of Hermite class and has a zero w, then the entirefunction

E(z)/(z w)ofz is of Hermite class. A sequence of polynomials Pn(z) exists such that

E(z)/Pn(z)

is an entire function of Hermite class for every nonnegative integer n and such that

E(z) = lim Pn(z)En(z)

uniformly on compact subsets of the upper halfplane for entire functions En(z) of Hermiteclass which have no zeros.

An analytic weight function is defined as a function W(z) of z which is analytic andwithout zeros in the upper halfplane. An entire function of Hermite class is an analyticweight function in the upper halfplane. Hilbert spaces of functions analytic in the upperhalfplane were introduced in Fourier analysis by Hardy.

The weighted Hardy spaceF(W) is defined as the Hilbert space of functions F(z) ofz ,which are analytic in the upper halfplane, such that the least upper bound

F2F(W) = sup

+

|F(x + iy)/W(x + iy)|2dx

taken over all positive y is finite. The classical Hardy space is obtained when W(z) isidentically one. Multiplication by W(z) is an isometric transformation of the classicalHardy space onto the weighted Hardy space with analytic weight function W(z).

An isometric transformation of the weighted Hardy space F(W) into itself is defined bytaking a function F(z) ofz into the function

F(z)(z w)/(z w)

of z when w is in the upper halfplane. The range of the transformation is the set ofelements of the space which vanish at w.

A continuous linear functional on the weighted Hardy spaceF(W) is defined by takinga function F(z) ofz into its value F(w) at w wheneverw is in the upper halfplane. Thefunction

W(z)W(w)/[2i(w z)]


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A PROOF OF THE RIEMANN HYPOTHESIS 3

ofz belongs to the space when w is in the upper halfplane and acts as reproducing kernelfunction for function values at w.

A Hilbert space of functions analytic in the upper halfplane which has dimensiongreater than one is isometrically equal to a weighted Hardy space if an isometric transfor-mation of the space onto the subspace of functions which vanish at w is defined by takingF(z) into

F(z)(z w)/(z w)when w is in the upper halfplane and if a continuous linear functional is defined on thespace by taking F(z) into F(w) forw of the upper halfplane.

Examples of weighted Hardy spaces in Fourier analysis are constructed from the Eulergamma function. The gamma function is a function (s) of s which is analytic in thecomplex plane with the exception of singularities at the nonpositive integers and whichsatisfies the recurrence relation

s(s) = (s + 1).

An analytic weight functionW(z) = (s)

is defined by

s= 12 iz.A maximal dissipative transformation is defined in the weighted Hardy space F(W) bytakingF(z) into F(z+ i) whenever the functions ofz belong to the space.

A relation T with domain and range in a Hilbert space is said to be accretive if thetransformation

(T

)/(T+ )

with domain and range in the Hilbert space is contractive for some, and hence every,complex number in the right halfplane. The relation Tis said to be maximal accretiveif the domain of the contractive transformation is the whole space for some, and henceevery, complex number in the right halfplane.

Theorem 1. A maximal accretive transformation is defined in a weighted Hardy spaceF(W) by takingF(z) into F(z + i) whenever the functions ofz belong to the space if, andonly if, the function

W(z 12 i)/W(z+ 12 i)

of z admits an extension which is analytic and has nonnegative real part in the upperhalfplane.

Proof of Theorem 1. A Hilbert spaceH whose elements are functions analytic in theupper halfplane is constructed when a maximal dissipative transformation is defined inthe weighted Hardy spaceF(W) by taking F(z) into F(z+i) whenever the functions ofz belong to the space. The spaceH is constructed from the graph of the adjoint of the


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transformation which takes F(z) into F(z+i) whenever the functions ofz belong to thespace.

An elementF(z) = (F+(z), F(z))

of the graph is a pair of analytic functions ofz, which belong to the spaceF(W), suchthat the adjoint takes F+(z) into F(z). The scalar product

F(t), G(t) = F+(t), G(t)F(W)+ F(t), G+(t)F(W)

of elements F(z) and G(z) of the graph is defined formally as the sum of scalar productsin the spaceF(W). Scalar selfproducts are nonnegative in the graph since the adjoint ofa maximal dissipative transformation is dissipative.

An element K(w, z) of the graph is defined by

K+(w, z) = W(z)W(w 12 i)/[2i(w + 12 i z)]

and

K(w, z) =W(z)W(w+ 12 i)

/[2i(w 12 i z)]when w is in the halfplane

1< iw iw.The identity

F+(w+ 12 i) + F(w 12 i) = F(t), K(w, t)

holds for every element

F(z) = (F+(z), F(z))

of the graph. An element of the graph which is orthogonal to itself is orthogonal to everyelement of the graph.

An isometric transformation of the graph onto a dense subspace ofH is defined bytaking

F(z) = (F+(z), F(z))

into the functionF+(z+

12 i) + F(z 12 i)

ofz in the halfplane

1< iz iz.The reproducing kernel function for function values at w in the spaceHis the function

[W(z+ 12 i)W(w 12 i) + W(z 12 i)W(w+ 12 i)]/[2i(w z)]

ofz in the halfplane when w is in the halfplane.


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Division by W(z+ 12 i) is an isometric transformation of the spaceH onto a Hilbertspace appearing in the Poisson representation of functions which are analytic and havenonnegative real part in the upper halfplane [1]. The function

(z) =W(z 12 i)/W(z+ 12 i)

ofz admits an analytic extension with nonnegative real part to the upper halfplane. The

function[(z) + (w)]/[2i(w z)]

of z belongs to the space when w is in the upper halfplane and acts as reproducingkernel function for function values at w. Since multiplication byW(z + 12 i) is an isometrictransformation of the space ontoH, the elements ofH have analytic extensions to theupper halfplane. The function


ofz belongs to the space when w is in the upper halfplane and acts as reproducing kernel

function for function values at w.The argument is reversed to construct a maximal accretive transformation in the weighted

Hardy space F(W) when the function(z) ofz admits an extension which is analytic andhas nonnegative real part in the upper halfplane. The Poisson representation constructsa Hilbert space whose elements are functions analytic in the upper halfplane and whichcontains the function

[(z) + (w)]/[2i(w z)]ofz as reproducing kernel function for function values at w when w is in the upper halfplane. Multiplication byW(z + 12 i) acts as an isometric transformation of the space onto aHilbert space

Hwhose elements are functions analytic in the upper halfplane and which

contains the function


ofz as reproducing kernel function for function values at w when w is in the upper halfplane.

A transformation is defined in the spaceF(W) by taking F(z) into F(z+ i) wheneverthe functions ofz belong to the space. The graph of the adjoint is a space of pairs

F(z) = (F+(z), F(z))

of elements of the space such that the adjoint takes the function F+(z) ofzinto the functionF(z) ofz . The graph contains

K(w, z) = (K+(w, z), K(w, z))

withK+(w, z) = W(z)W(w 12 i)/[2i(w + 12 i z)]


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andK(w, z) =W(z)W(w+

12 i)

/[2i(w 12 i z)]when w is in the halfplane

1< iw iw.The elements K(w, z) of the graph span the graph of a restriction of the adjoint. The

transformation in the spaceF(W) is recovered as the adjoint of the restricted adjoint.A scalar product is defined on the graph of the restricted adjoint so that an isometric

transformation of the graph of the restricted adjoint into the space His defined by taking

F(z) = (F+(z), F(z))

intoF+(z+

12 i) + F(z 12 i).

The identity

F(t), G(t)

=F+(t), G(t)

F(W)+

F(t), G+(t)

F(W)

holds for all elementsF(z) = (F+(z), F(z))

andG(z) = (G+(z), G(z))

of the graph of the restricted adjoint. The restricted adjoint is accretive since scalar selfproducts are nonnegative in its graph. The adjoint is accretive since the transformationin the spaceF(W) is the adjoint of its restricted adjoint.

The accretive property of the adjoint is expressed in the inequality

F+(t) F(t)F(W) F+(t) + F(t)F(W)for elements

F(z) = (F+(z), F(z))

of the graph when is in the right halfplane. The domain of the contractive transforma-tion which takes the function

F+(z) + F(z)

ofz into the functionF+(z)

F(z)

ofz is a closed subspace of the space F(W). The maximal accretive property of the adjointis the requirement that the contractive transformation be everywhere defined for some, andhence every,in the right halfplane.

SinceK(w, z) belongs to the graph whenw is in the halfplane

1< iw iw,


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an elementH(z) of the spaceF(W) which is orthogonal to the domain satisfies the identity

H(w 12 i) + H(w+ 12 i) = 0

when w is in the upper halfplane. The function H(z) ofz admits an analytic extensionto the complex plane which satisfies the identity

H(z) + H(z+ i) = 0.

A zero ofH(z) is repeated with periodi. Since

H(z)/W(z)

is analytic and of bounded type in the upper halfplane, the function H(z) ofz vanisheseverywhere if it vanishes somewhere.

The space of elements H(z) of the spaceF(W) which are solutions of the equation

H(z) + H(z+ i) = 0

for somein the right halfplane has dimension zero or one. The dimension is independentof.

If is positive, multiplication byexp(i z)

is an isometric transformation of the spaceF(W) into itself which takes solutions of theequation for a given into solutions of the equation with replaced by

exp().

A solutionH(z) of the equation for a given vanishes identically since the function

exp(i z)H(z)

ofz belongs to the space for every positive number and has the same norm as the functionH(z) ofz.

The transformation which takes F(z) into F(z+ i) whenever the functions ofz belongto the spaceF(W) is maximal dissipative since it is the adjoint of its adjoint, which ismaximal dissipative.

This completes the proof of the theorem.

An example of an analytic weight function which satisfies the hypotheses of the theoremis obtained when

W(z) = ( 12 iz)since

W(z+ 12 i)/W(z 12 i) = iz


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is analytic and has nonnegative real part in the upper halfplane by the recurrence relationfor the gamma function. The weight functions which satisfy the hypotheses of the theoremare generalizations of the gamma function in which an identity is replaced by an inequality.

The theorem has another formulation. A maximal accretive transformation is definedin a weighted Hardy spaceF(W) for some real number h by taking F(z) into F(z+ih)whenever the functions ofz belong to the space if, and only if, the function

W(z+ 12 ih)/W(z 12 ih)

of z admits an extension which is analytic and has nonnegative real part in the upperhalfplane.

Another theorem is obtained in the limit of small h. A maximal accretive transformationis defined in a weighted Hardy spaceF(W) by taking F(z) into iF(z) whenever thefunctions ofz belong to the space if, and only if, the function

iW(z)/W(z)

ofz has nonnegative real part in the upper halfplane.

The proof of the theorem is similar to the proof of Theorem 1. A relationship to theHermite class of entire functions is indicated in another formulation. A maximal accretivetransformation is defined in a weighted Hardy space F(W) by taking F(z) into iF(z)whenever the functions ofz belong to the space if, and only if, the modulus ofW(x + iy)is a nondecreasing function of positive y for every real numberx.

An Euler weight function is defined as an analytic weight function W(z) such that amaximal accretive transformation is defined in the weighted Hardy space F(W) wheneverhis in the interval [1, 1] by takingF(z) intoF(z +ih) whenever the functions ofz belongto the space.

If a nontrivial function(z) ofz is analytic and has nonnegative real part in the upperhalfplane, a logarithm of the functions is defined continuously in the halfplane withvalues in the strip of width centered on the real line. The inequalities

i log (z) i log (z)

are satisfied. A function h(z) ofz which is analytic and has nonnegative real part in theupper halfplane is defined whenhis in the interval (1, 1) by the integral

log h(z) = sin(h) +

log (z

t)dt

cos(2it) + cos(h) .

An application of the Cauchy formula in the upper halfplane shows that the function

sin(h)

cos(2iz) + cos(h)=

+

exp(2itz) exp(ht exp(ht)

exp(t) exp(t) dt


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ofz is the Fourier transform of a function

exp(ht) exp(ht)exp(t) exp(t)

of positivet which is square integrable with respect to Lebesgue measure and bounded byh when h is in the interval (0, 1).

The identityh(z) = h(z)

1

is satisfied. The function(z) = lim h(z)

ofz is recovered in the limit as h increases to one. The identity

a+b(z) =a(z 12 ib)b(z+ 12 ia)

when a, b, and a+b belong to the interval (1, 1) is a consequence of the trigonometricidentity

sin(a + b)

cos(2iz) + cos(a + b)

= sin(a)

cos(2iz+ b) + cos(a)+

sin(b)

cos(2iz a) + cos(b) .

An Euler weight functionW(z) is defined within a constant factor by the limit

iW(z)/W(z) = lim log h(z)

h =

+

log (z t)dt1 + cos(2it)

.

The identityW(z+ 12 ih) = W(z 12 ih)h(z)

applies whenh is in the interval (1, 1). The identity reads

W(z+ 12 i) = W(z 12 i)(z)

in the limit as h increases to one.

An Euler weight functionW(z) has been constructed which satisfies the identity

W(z+ 12 i) = W(z 12 i)(z)

for a given nontrivial function (z) ofz which is analytic and has nonnegative real part inthe upper halfplane.

If a maximal accretive transformation is defined in a weighted Hardy spaceF(W) bytakingF(z) intoF(z +i) whenever the functions ofz belong to the space, then the identity

W(z+ 12 i) = W(z 12 i)(z)


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holds for a function(z) ofz which is analytic and has nonnegative real part in the upperhalfplane. The analytic weight function W(z) is the product of an Euler weight functionand an entire function which is periodic of period iand has no zeros.

IfW(z) is an Euler weight function, the maximal accretive transformation defined for hin the interval [0, 1] by takingF(z) intoF(z +ih) whenever the functions ofz belong to thespace

F(W) is subnormal: The transformation is the restriction to an invariant subspace

of a normal transformation in the larger Hilbert spaceHof (equivalence classes of) BairefunctionsF(x) of realx for which the integral

F2 = +

|F(t)/W(t)|2dt

converges. The passage to boundary value functions maps the spaceF(W) isometricallyinto the spaceH.

For givenh the function

(z) = W(z+ 12 ih)/W(z 12 ih)ofz is analytic and has nonnegative real part in the upper halfplane. The transformationT is defined to take F(z) into F(z+ih) whenever the functions ofz belong to the spaceF(W). The transformation Sis defined to take F(z) into

(z+ 12 ih)F(z)

whenever the functions ofz belong to the spaceF(W). The adjoint

T

=S

1

T S

ofTis computable from the adjointS ofSon a dense subset of the space F(W) containingthe reproducing kernel functions for function values. The transformation T1ST takesF(z) into

(z 12 ih)1F(z)on a dense set of elements F(z) of the spaceF(W). The transformation T1T is the re-striction of a contractive transformation of the space F(W) into itself. The transformationtakesF(z) intoG(z) when the boundary value functionG(x) is the orthogonal projectionof

F(x)(x 1

2 ih)

/(x 1

2 ih)in the image of the spaceF(W).

An isometric transformation of the spaceH onto itself is defined by taking a functionF(x) of real x into the function

F(x)(x 12 ih)/(x 12 ih)


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of realx. A dense set of elements ofHare productsexp(iax)F(x)

for nonnegative numbersa withF(x) the boundary value function of a function F(z) ofzwhich belongs to the spaceF(W). A normal transformation is defined in the spaceH asthe closure of a transformation which is computed on such elements. The transformation

takes the functionexp(iax)F(x)

of realxinto the functionexp(iah)exp(iax)G(x)

of realx for every nonnegative number awhen F(z) and

G(z) =F(z+ ih)

are functions ofz in the upper halfplane which belong to the spaceF(W).Entire functions of Hermite class are examples of analytic weight functions which are

limits of polynomials having no zeros in the upper halfplane. Such polynomials appearin the Stieltjes representation of positive linear functionals on polynomials.

A linear functional on polynomials with complex coefficients is said to be nonnegativeif it has nonnegative values on polynomials whose values on the real axis are nonnegative.A positive linear functional on polynomials is a nonnegative linear functional on polyno-mials which does not vanish identically. A nonnegative linear functional on polynomials isrepresented as an integral with respect to a nonnegative measure on the Baire subsetsof the real line. The linear functional takes a polynomial F(z) into the integral

F(t)d(t).

Stieltjes examines the action of a positive linear functional on polynomials of degree lessthanr for a positive integerr. A polynomial which as nonnegative values on the real axisis a product

F(z)F(z)

of a polynomial F(z) and the conjugate polynomial

F(z) =F(z).

If the positive linear functional does not annihilateF(z)F(z)

for any nontrivial polynomialF(z) of degree less than r, then a Hilbert space exists whoseelements are the polynomials of degree less than r and whose scalar product

F(t), G(t)


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is defined as the action of the positive linear functional on the polynomial

G(z)F(z).

Stieltjes shows that the Hilbert space of polynomials of degree less than r is containedisometrically in a weighted Hardy spaceF(W) whose analytic weight function W(z) is apolynomial of degree r having no zeros in the upper halfplane.

Examples of such spaces are applied by Legendre in quadratic approximations of periodicmotion and motivate the application to number theory made precise by the Riemannhypothesis.

An axiomatization of the Stieltjes spaces is stated in a general context [2]. Hilbertspaces are examined whose elements are entire functions and which have these properties:

(H1) Whenever an entire function F(z) ofz belongs to the space and has a nonreal zerow, the entire function

F(z)(z w)/(z w)

ofz belongs to the space and has the same norm as F(z).(H2) A continuous linear functional on the space is defined by taking a function F(z)

ofz into its value F(w) at w for every nonreal number w.

(H3) The entire functionF(z) =F(z)

ofz belongs to the space whenever the entire functionF(z) ofz belongs to the space, andit has the same norm as F(z).

An example of a Hilbert space of entire functions which satisfies the axioms is obtainedwhen an entire function E(z) ofz satisfies the inequality

|E(x y)| < |E(x + iy)|

for all real x when y is positive. A weighted Hardy spaceF(W) is defined with analyticweight function

W(z) =E(z).

A Hilbert spaceH(E) which is contained isometrically in the spaceF(W) is defined asthe set of entire functions F(z) ofz such that the entire functions F(z) ofz and F(z) ofz belong to the spaceF(W). The entire function

[E

(z

)E

(w

)

E(

z)E

(w

)]/

[2i

(w

z)]

ofz belongs to the space H(E) for every complex numberw and acts as reproducing kernelfunction for function values at w.

A Hilbert spaceH of entire functions which satisfies the axioms (H1), (H2), and (H3)is isometrically equal to a spaceH(E) if it contains a nonzero element. The proof appliesreproducing kernel functions which exist by the axiom (H2).


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For every nonreal numberw a unique entire function K(w, z) ofz exists which belongsto the space and acts as reproducing kernel function for function values atw. The functiondoes not vanish identically since the axiom (H1) implies that some element of the spacehas a nonzero value at w when some element of the space does not vanish identically.The scalar selfproductK(w, w) of the function K(w, z) ofz is positive. The axiom (H3)implies the symmetry

K(w

, z) =K(w, z

)

.

If is a nonreal number, the set of elements of the space which vanish at is a Hilbertspace of entire functions which is contained isometrically in the given space. The function

K(w, z) K(w, )K(, )1K(, z)

ofz belongs to the subspace and acts as reproducing kernel function for function values at. The identity

[K(w, z) K(w, )K(, )1

K(, z)](z

)(w

)= [K(w, z) K(w, )K(, )1K(, z)](z )(w )

is a consequence of the axiom (H1).

An entire function E(z) ofz exists such that the identity

K(w, z) = [E(z)E(w) E(z)E(w)]/[2i(w z)]

holds for all complex z when w is not real. The entire function can be chosen with azero at when is in the lower halfplane and with a zero at when is in the upper

halfplane. The function is then unique within a constant factor of absolute value one. AspaceH(E) exists and is isometrically equal to the given spaceH.

Examples of Hilbert spaces of entire functions which satisfy the axioms (H1), (H2), and(H3) are constructed from the analytic weight function

W(z) = aiz(12 iz)

for every positive number a. The space is contained isometrically in the weighted HardyspaceF(W) and contains every entire function F(z) such that the functions F(z) andF(z) ofz belong to the space

F(W). The space of entire functions is isometrically equal

to a spaceH(E) whose defining function E(z) is a confluent hypergeometric series [1].Properties of the space define a class of Hilbert spaces of entire functions.

An Euler space of entire functions is a Hilbert space of entire functions which satisfiesthe axioms (H1), (H2), and (H3) such that a maximal accretive transformation is definedin the space for every h in the interval [1, 1] by takingF(z) intoF(z+ ih) whenever thefunctions ofz belong to the space.


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Theorem 2. A maximal accretive transformation is defined in a Hilbert spaceH(E) ofentire functions for a real numberh by takingF(z) into F(z+ ih) whenever the functionsofz belong to the space if, and only if, a Hilbert spaceH of entire functions exists whichcontains the function

[E(z+ 12 ih)E(w 12 ih) E(z+ 12 ih)E(w + 12 ih)]/[2i(w z)]

+ [E(z 12 ih)E(w+

12 ih)

E

(z 12 ih)E(w

12 ih)]/[2i(w

z)]ofz as reproducing kernel function for function values atw for every complex numberw.

Proof of Theorem 2. The spaceH is constructed from the graph of the adjoint of thetransformation which takes F(z) intoF(z+ ih) whenever the functions ofz belong to thespace. An element

F(z) = (F+(z), F(z))

of the graph is a pair of entire functions ofz , which belong to the spaceH(E), such thatthe adjoint takes F+(z) into F(z). The scalar product

F(t), G(t) = F+(t), G(t)H(E)+ F(t), G+(t)H(E)

of elements F(z) and G(z) of the graph is defined as a sum of scalar products in thespace H(E). Scalar selfproducts are nonnegative since the adjoint of a maximal accretivetransformation is accretive.

An elementK(w, z) = (K+(w, z), K(w, z))

of the graph is defined for every complex number w by

K+(w, z) = [E(z)E(w 12 ih) E(z)E(w + 12 ih)]/[2i(w + 12 ih z)]

and

K(w, z) = [E(z)E(w+ 12 ih)

E(z)E(w 12 ih)]/[2i(w 12 ih z)].

The identityF+(w+

12 ih) + F(w 12 ih) = F(t), K(w, t)

holds for every element

F(z) = (F+(z), F(z))of the graph. An element of the graph which is orthogonal to itself is orthogonal to everyelement of the graph.

A partially isometric transformation of the graph onto a dense subspace of the space His defined by taking

F(z) = (F+(z), F(z))


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into the entire functionF+(z+

12 ih) + F(z 12 ih)

ofz. The reproducing kernel function for function values at win the space H is the function

[E(z+ 12 ih)E(w 12 ih) E(z+ 12 ih)E(w + 12 ih)]/[2i(w z)]+ [E(z

12 ih)E(w+

12 ih)

E(z

12 ih)E(w

12 ih)]/[2i(w

z)]

ofz for every complex number w.

This completes the construction of a Hilbert space H of entire functions with the desiredreproducing kernel functions when the maximal accretive transformation exists in the spaceH(E). The argument is reversed to construct the maximal accretive transformation in thespace H(E) when the Hilbert space of entire functions with the desired reproducing kernelfunctions exists.

A transformation is defined in the spaceH(E) by taking F(z) intoF(z+ ih) wheneverthe functions ofz belong to the space. The graph of the adjoint is a space of pairs

F(z) = (F+(z), F(z))

such that the adjoint takes the function F+(z) ofz into the functionF(z) ofz. The graphcontains

K(w, z) = (K+(w, z), K(w, z))

with

K+(w, z) = [E(z)E(w 12 ih) E(z)E(w + 12 ih)]/[2i(w + 12 ih z)]

and

K(w, z) = [E(z)E(w+ 12 ih)

E(z)E(w 12 ih)]/[2i(w 12 ih z)]

for every complex number w. The elements K(w, z) of the graph span the graph of arestriction of the adjoint. The transformation in the space H(E) is recovered as the adjointof its restricted adjoint.

A scalar product is defined on the graph of the restricted adjoint so that an isometrictransformation of the graph of the restricted adjoint into the space His defined by taking

F(z) = (F+(z), F(z))

intoF+(z+

12 ih) + F(z 12 ih).

The identityF(t), G(t) = F+(t), G(t)H(E)+ F(t), G+(t)H(E)

holds for all elementsF(z) = (F+(z), F(z))


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of the graph of the restricted adjoint. The restricted adjoint is accretive since scalar selfproducts are nonnegative in its graph. The adjoint is accretive since the transformationin the spaceH(E) is the adjoint of its restricted adjoint.

The accretive property of the adjoint is expressed in the inequality

F+(t) F(t)H(E) F+(t) + F(t)H(E)for elements

F(z) = (F+(z), F(z))

of the graph when is in the right halfplane. The domain of the contractive transforma-tion which takes the function

F+(z) + F(z)

ofz into the functionF+(z) F(z)

ofz is a closed subspace of the space H(E). The maximal accretive property of the adjointis the requirement that the contractive transformation be everywhere defined for some,and hence every,in the right halfplane.

Since K(w, z) belongs to the graph for every complex number w, an entire functionH(z) ofz which belongs to the spaceH(E) and is orthogonal to the domain is a solutionof the equation

H(z) + H(z+ i) = 0.

The function vanishes identically if it has a zero since zeros are repeated periodically withperiodi and since the function

H(z)/E(z)

ofz is of bounded type in the upper halfplane. The space of solutions has dimension zero

or one. The dimension is zero since it is independent of.

The transformation which takesF(z) intoF(z + ih) whenever the functions ofz belongto the spaceH(E) is maximal accretive since it is the adjoint of its adjoint, which ismaximal accretive.


The defining functionE(z) of an Euler space of entire functions is of Hermite class sincethe function

E(z 12 ih)/E(z+ 12 ih)ofz is of bounded type and of nonpositive mean type in the upper halfplane when h isin the interval (0, 1). Since the function is bounded by one on the real axis, it is boundedby one in the upper halfplane. The modulus ofE(x + iy) is a nondecreasing function ofpositivey for every real x. An entire functionF(z) ofz which belongs to the spaceH(E)is of Hermite class if it has no zeros in the upper halfplane and if the inequality

|F(x iy)| |F(x + iy)|


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holds for all realx when y is positive.

A construction of Hilbert spaces of entire functions which satisfy the axioms (H1), (H2),and (H3) appears in the spectral theory of ordinary differential equations of second orderwhich are formally selfadjoint. The spectral theory is advantageously reformulated asa spectral theory of first order differential equations for pairs of scalar functions. Theresulting canonical form is the classical model for a vibrating string.

The canonical form for the integral equation is obtained with a continuous function ofpositivet whose values are matrices

m(t) =

(t) (t)(t) (t)

with real entries such that the matrix inequality

m(a) m(b)

holds when ais less than b. It is assumed that (t) is positive when t is positive, that

lim (t) = 0

as t decreases to zero, and that the integral

10

(t)d(t)

is finite.

The matrixI=

0 11 0

is applied in the formulation of the integral equation. When a is positive, the integralequation

M(a,b,z)I I=z ba

M(a,t,z)dm(t)

admits a unique continuous solution

M(a,b,z) = A(a,b,z) B(a,b,z)C(a,b,z) D(a,b,z)

as a function ofb greater than or equal to a for every complex number z. The entries ofthe matrix are entire functions ofz which are selfconjugate and of Hermite class for everyb. The matrix has determinant one. The identity

M(a,c,z) =M(a,b,z)M(b,c,z)


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holds when a b c.A bar is used to denote the conjugate transpose

M =

A C

B D

of a square matrixM=

A BC D

with complex entries and also for the conjugate transpose

c = (c+, c

)

of a column vector

c=

c+c

with complex entries. The space of column vectors with complex entries is a Hilbert spaceof dimension two with scalar product

u, v =vu= v+u++ vu.

When a and b are positive with a less than or equal to b, a unique Hilbert spaceH(M(a, b)) exists whose elements are pairs

F(z) =

F+(z)F(z)

of entire functions ofz such that a continuous transformation of the space into the Hilbertspace of column vectors is defined by taking F(z) into F(w) for every complex number wand such that the adjoint takes a column vector c into the element

[M(a,b,z)IM(a,b,w) I]c/[2(z w)]

of the space.

An entire functionE(c, z) = A(c, z) iB(c, z)

ofz which is of Hermite class exists for every positive number csuch that the selfconjugate

entire functions A(c, z) and B(c, z) satisfy the identity

(A(b, z), B(b, z)) = (A(a, z), B(a, z))M(a,b,z)

when a is less than or equal to b and such that the entire functions

E(c, z)exp[(c)z]


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ofz converge to one uniformly on compact subsets of the complex plane as c decreases tozero.

A spaceH(E(c)) exists for every positive number c. The spaceH(E(a)) is containedcontractively in the spaceH(E(b)) when a is less than or equal to b. The inclusion isisometric on the orthogonal complement in the spaceH(E(a)) of the elements which arelinear combinations

A(a, z)u + B(a, z)v

with complex coefficients u and v. These elements form a space of dimension zero or onesince the identity

vu= uv

is satisfied.

A positive number b is said to be singular with respect to the function m(t) of t if itbelongs to an interval (a, c) such that equality holds in the inequality

[(c) (a)]2 [(c) (a)][(c) (a)]

with m(b) unequal to m(a) and unequal to m(c). A positive number is said to be regularwith respect to m(t) if it is not singular with respect to the function oft.

Ifaandcare positive numbers such thatais less thancand if an elementbof the interval(a, c) is regular with respect to m(t), then the space H(M(a, b)) is contained isometricallyin the spaceH(M(a, c)) and multiplication by M(a,b,z) is an isometric transformationof the spaceH(M(b, c)) onto the orthogonal complement of the spaceH(M(a, b)) in thespaceH(M(a, c)).

Ifa and b are positive numbers such that a is less than b and ifa is regular with respectto m(t), then the spaceH(E(a)) is contained isometrically in the spaceH(E(b)) and anisometric transformation of the spaceH(M(a, b)) onto the orthogonal complement of thespaceH(E(a)) in the spaceH(E(b)) is defined by taking

F+(z)F(z)

into 2 [A(a, z)F+(z) + B(a, z)F(z)].

A function(t) of positive twith real values exists such that the function

m(t) + Iih(t)of positive t with matrix values is nondecreasing for a function h(t) oft with real valuesif, and only if, the functions

(t) h(t)and

(t) + h(t)


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of positivet with real values are nondecreasing. The function (t) oft, which is continuousand nondecreasing, is called a greatest nondecreasing function such that

m(t) + Ii(t)

is nondecreasing. The function is unique within an added constant.

Ifaand b are positive numbers such that ais less than b, multiplication by

exp(ihz)

is a contractive transformation of the spaceH(E(a)) into the spaceH(E(b)) for a realnumber h, if, and only if, the inequalities

(a) (b) h (b) (a)

are satisfied. The transformation is isometric when a is regular with respect to m(t).An analytic weight function W(z) may exist such that multiplication by

exp(i(c)z)

is an isometric transformation of the spaceH(E(c)) into the weighted Hardy spaceF(W)for every positive number c which is regular with respect to m(t). The analytic weightfunction is unique within a constant factor of absolute value one if the function

(t) + (t)

of positive t is unbounded in the limit of large t. The function

W(z) = lim E(c, z)exp(i(c)z)

can be chosen as a limit uniformly on compact subsets of the upper halfplane.

If multiplication by

exp(i z)

is an isometric transformation of a spaceH

(E) into the weighted Hardy spaceF

(W) forsome real number and if the spaceH(E) contains an entire function F(z) wheneverits product with a nonconstant polynomial belongs to the space, then the spaceH(E) isisometrically equal to the spaceH(E(c)) for some positive numberc which is regular withrespect to m(t).

A construction of Euler spaces of entire functions is made from Euler weight functionswhen a hypothesis is satisfied.


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Theorem 3. If for some real number a nontrivial entire functionF(z) ofz exists suchthat the functions

exp(i z)F(z)

andexp(iz)F(z)

ofz belong to the weighted Hardy spaceF(W) of an Euler weight functionW(z), then anEuler space of entire functions exists such that multiplication byexp(i z) is an isometrictransformation of the space into the weighted Hardy space and such that the space containsevery entire functionF(z) ofz such that the functions

exp(i z)F(z)

andexp(iz)F(z)

ofz belong to the weighted Hardy space.

Proof of Theorem 3. The set of entire functions F(z) such that the functions

exp(i z)F(z)

andexp(iz)F(z)

ofzbelong to the weighted Hardy space is a vector space with scalar product determined bythe isometric property of multiplication as a transformation of the space into the weightedHardy space. The space is shown to be a Hilbert space by showing that a Cauchy sequence

of elements Fn(z) of the space converge to an element F(z) of the space.

Since the elementsexp(i z)Fn(z)

andexp(iz)Fn(z)

of the weighted Hardy space form Cauchy sequences, a function F(z) ofz which is analyticseparately in the upper halfplane and the lower halfplane exists such that the limitfunctions

exp(i z)F(z) = lim exp(i z)Fn(z)

andexp(iz)F(z) = lim exp(i z)Fn(z)

ofz belong to the weighted Hardy space. Since

|z z| 12 exp(i z)F(z)/W(z) = lim |z z| 12 exp(i z)Fn(z)/W(z)


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and|z z| 12 exp(i z)F(z)/W(z) = lim |z z| 12 exp(i z)Fn(z)/W(z)

uniformly in the upper halfplane and since the functions

log |Fn(z)/W(z)|

andlog |Fn(z)/W(z)|

ofz are subharmonic in the halfplane

1< iz iz,

the convergence ofF(z) = lim Fn(z)

is uniform on compact subsets of the complex plane. The limit function F(z) of z isanalytic in the complex plane.

A Hilbert space of entire functions which satisfies the axioms (H1), (H2), and (H3) isobtained such that multiplication by exp(i z) is an isometric transformation of the spaceinto the weighted Hardy space. Since the space contains a nonzero element by hypothesis,it is isometrically equal to a spaceH(E).

The space is shown to be an Euler space of entire functions by showing that a maximalaccretive transformation is defined in the space forh in the interval [1, 1] by takingF(z)intoF(z+ ih) whenever the functions ofz belong to the space. The accretive property ofthe transformation is a consequence of the accretive property in the weighted Hardy space.

Maximality is proved by showing that every element of the space is a sum

F(z) + F(z+ ih)

of functionsF(z) and F(z+ ih) ofz which belong to the space.

Since a maximal accretive transformation exists in the weighted Hardy space, everyelement of the Hilbert space of entire functions is in the upper halfplane a sum

F(z) + F(z+ ih)

of functionsF(z) and F(z+ ih) ofz such that the functions

exp(i z)F(z)

andexp(i z)F(z+ ih)

of z belong to the weighted Hardy space. The function F(z) of z admits an analyticcontinuation to the complex plane. The decomposition applies for all complex z.


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The entire functionF(z) + F(z ih)

of z belongs to the Hilbert space of entire functions since the space satisfies the axiom(H3). An entire function G(z) ofz exists such that

F(z) + F(z

ih) = G(z) + G(z+ ih)

and such that the functionsexp(i z)G(z)

andexp(i z)G(z+ ih)

ofz belong to the weighted Hardy space.

Vanishing of the entire function

F(z)

G(z+ ih) = G(z)

F(z

ih)

of z implies the desired conclusion that the functions F(z) and F(z + ih) of z as wellas the functions G(z) and G(z+ih) ofz belong to the Hilbert space of entire functions.Vanishing is proved by showing boundedness of the function in the strip

2h < iz iz


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apply when z is in the strip, the inequality

|F(z) G(z+ ih)|2 | exp(i z)W(z)|2/(iz iz)+ exp(2h)| exp(i z)W(z+ ih)|2/(2h + iz iz)

applies whenz is in the strip.

Boundedness of the entire function

F(z) G(z+ ih)

ofz in the complex plane follows from the subharmonic property of the logarithm of itsmodulus. The entire function is a constant which vanishes because of the identity

F(z) G(z+ ih) =G(z) F(z ih).


The hypothesis of the theorem are satisfied by an Euler weight function W(z) whichsatisfies the identity

W(z+ 12 i) =W(z 12 i)(z)for a function (z) of z which is analytic and has nonnegative real part in the upperhalfplane if log (z) has nonnegative real part in the upper halfplane. The modulus ofW(x + iy) is then a nondecreasing function of positivey for every realx.

Since the weight function can be multiplied by a constant, it can be assumed to havevalue one at the origin. The phase (x) is defined as the continuous function of real x

with value zero at the origin such that

exp(i(x))W(x)

is positive for all real x. The phase function is a nondecreasing function of real x which isidentically zero if it is constant in any interval.

When the phase function vanishes identically, the modulus ofW(x+iy) is a constantas a function of positive y for every real x. The weight function is then the restriction ofa selfconjugate entire function of Hermite class. For every positive number an entirefunction

F(z) =W(z)sin( z)/z

is obtained such that the functions

exp(i z)F(z)

andexp(iz)F(z)


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of z belong to the spaceF(W). No nonzero entire function F(z) exists such that thefunctionsF(z) andF(z) belong to the spaceF(W).

When the phase function does not vanish identically, an entire function E(z) of Hermiteclass which has no real zeros exists such that E(x) is real for a real number x if, and onlyif,(x) is an integral is an integral multiple of, and then

exp(i(x))E(x)

is positive. Such an entire function is unique within a factor of a selfconjugate entirefunction of Hermite class. The factor is chosen so that the function

E(z)/W(z)

of z has nonnegative real part in the upper halfplane. The entire functions E(z) andE(z) are linearly independent. A nontrivial entire function

F(z) = [E(z) E(z)]/z

is obtained such that the functions F(z) andF(z) ofz belong to the spaceF(W).The same conclusions are obtained under a weaker hypothesis.

Theorem 4. If an Euler weight functionW(z) satisfies the identity

W(z+ 12 i) = W(z 12 i)(z)

for a function(z) ofz which is analytic and has nonnegative real part in the upper halfplane such that

(z) + log (z)

has nonnegative real part in the upper halfplane for a function(z)ofz which is analyticand has nonnegative real part in the upper halfplane such that the least upper bound

sup

+

R(x + iy)dx

taken over all positive y is finite, then for every positive number a nontrivial entirefunctionF(z) exists such that the functions

exp(i z)F(z)

and exp(iz)F(z)

ofz belong to the weighted Hardy spaceF(W).

Proof of Theorem 4. It can be assumed that the symmetry condition

(z) = (z)


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is satisfied since otherwise(z) can be replaced by(z)+(z). Whenhis in the interval(0, 1), the integral

log h(z) = sin(h)

+

log (z t)dtcos(2it) + cos(h)

defines a function h(z) ofz which is analytic and has nonnegative real part in the upperhalfplane such that

W(z+ 12 ih) = W(z 12 ih)h(z).The integral

Rh(z) = sin(h) +

R(z t)dtcos(2it) + cos(h)

and the symmetry conditionn(z) =n(z)

define a function h(z) which is analytic and has nonnegative real part in the upper halfplane. The function

h(z) + log h(z)

ofz has nonnegative real part in the upper halfplane since the function

(z) + log (z)

has nonnegative real part in the upper halfplane by hypothesis.

An analytic weight function U(z) which admits an analytic extension without zeros tothe halfplane iz iz > 1 is defined within a constant factor by the identity

log U(z+ 1

2

ih)

log U(z

1

2

ih) =h(z)

forh in the interval (0, 1) and by the symmetry

U(z) =U(z).

The analytic weight functionV(z) = U(z)W(z)

has an analytic extension without zeros to the halfplane iz iz > 1. The modulus ofU(x + iy) and the modulus ofV(x + iy) are nondecreasing functions of positive y for everyrealx.

Since

ylog |U(x + iy)| =

+

R(x + iy t)dt1 + cos(2it)

the integral +

ylog |U(x + iy)|dx=

+

R(x + iy)dx


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is a bounded function of positivey . The phase(x) is the continuous, nondecreasing, oddfunction of real x such that

exp(i(x))U(x)

is positive for all real x. Since

y log |U(x + iy)| = y

+

d(t)

(t x)2 + y2

when y is positive, the inequality

(b) (a) sup +

R(x + iy)dx

holds when ais less than b with the least upper bound taken over all positive y.

The remaining arbitrary constant in U(z) is chosen so that the integral representation

log U(z) = 12

0

log(1 z2/t2)d(t)

holds when z is in the upper halfplane with the logarithm of 1 z2/t2 defined continu-ously in the upper halfplane with nonnegative values when z is on the upper half of theimaginary axis. The inequality

|U(z)| |U(i|z|)holds when z is in the upper halfplane since

|1 z2/t2| 1 + zz/t2.

If a positive integer r is chosen so that the inequality

+

R(x + iy)dx 2r

holds for all positivey, then the function

U(z)/(z+ i)r

is bounded and analytic in the upper halfplane.

Since the modulus ofV(x +iy) is a nondecreasing function of positive y for every realx, there exists for every positive number a nontrivial entire functionG(z) such that thefunctions

exp(i z)G(z)

andexp(i z)G(z)


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ofz belong to the weighted Hardy spaceF(V). Since the entire function

G(z) = F(z)P(z)

is the product of an entire function F(z) and a polynomial P(z) of degree r, a nontrivialentire function F(z) is obtained such that the functions

exp(i z)F(z)

andexp(iz)F(z)

ofz belong to the weighted Hardy spaceF(W).This completes the proof of the theorem.

The Hilbert spaces of entire functions for the vibrating string of an Euler weight functionare Euler spaces of entire functions.

Theorem 5. A Hilbert space of entire functions which satisfies the axioms (H1), (H2),and (H3) and which contains a nonzero element is an Euler space of entire functions if itcontains an entire function whenever its product with a nonconstant polynomial belongs tothe space and if multiplication byexp(i z) is for some real numberan isometric trans-

formation of the space into the weighted Hardy spaceF(W) of an Euler weight functionW(z).

Proof of Theorem 5. It can be assumed that vanishes since the function

exp(

i z)W(z)

is an Euler weight function whenever the function W(z) ofz is an Euler weight function.

The given Hilbert space of entire functions is isometrically equal to a space H(E) for anentire function E(z) which has no real zeros since an entire function belongs to the spacewhenever its product with a nonconstant polynomial belongs to the space.

An accretive transformation is defined in the space H(E) whenh is in the interval [0, 1]by taking F(z) into F(z+ ih) whenever the functions ofz belong to the space since thespace is contained isometrically in the space F(W) and since an accretive transformation isdefined in the space F(W) by takingF(z) intoF(z+ih) whenever the functions ofz belongto the space. It remains to prove the maximal accretive property of the transformation inthe spaceH(E).

The ordering theorem for Hilbert spaces of entire functions applies to spaces whichsatisfy the axioms (H1), (H2), and (H3) and which are contained isometrically in a weightedHardy spaceF(W) when a space contains an entire function whenever its product witha nonconstant polynomial belongs to the space. One space is properly contained in theother when the two spaces are not identical.


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A Hilbert spaceH of entire functions which satisfies the axioms (H1) and (H2) andwhich contains a nonzero element need not satisfy the axiom (H3). Multiplication byexp(iaz) is for some real number a an isometric transformation of the space onto a Hilbertspace of entire functions which satisfies the axioms (H1), (H2), and (H3).

A space H which satisfies the axioms (H1) and (H2) and which is contained isometricallyin the space

F(W) is defined as the closure in the space

F(W) of the set of those elements

of the space which functions F(z+ ih) ofz for functionsF(z) ofz belonging to the spaceH(E). An example of a function F(z+ ih) ofz is obtained for every element of the spaceH(E) which is a functionF(z) ofz such that the functionz2F(z) ofz belongs to the spaceH(E). The spaceH contains an entire function whenever its product with a nonconstantpolynomial belongs to the space.

The functionE(z)/W(z)

ofz is of bounded type in the upper halfplane and has the same mean type as the function

E(z+ ih)/W(z+ ih)

ofz which is of bounded type in the upper halfplane. Since the function

W(z+ ih)/W(z)

ofz is of bounded type and has zero mean type in the upper halfplane, the function

E(z+ ih)/E(z)

ofz is of bounded type and of zero mean type in the upper halfplane.

If a function F(z) ofz is an element of the spaceH

(E) such that the functions

F(z)/W(z)

andF(z)/W(z)

ofz have equal mean type in the upper halfplane, and such that the functions

G(z) =F(z+ ih)

and

G

(z) =F

(z ih)ofz belong to the spaceF(W), then the functions

G(z)/W(z)

andG(z)/W(z)


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whenever the functions ofz belong to the space. If

E(z) =A(z) iB(z)

for selfconjugate entire functions A(z) and B(z), the identities

[A(z+ i) A(z)]/(1

2 iz) = A(z)s iB(z)rand

[B(z+ i) B(z)]/(12 iz) = iA(z)p + B(z)shold for positive numbers p, r, and s such that

pr= s2.

The spaces are parametrized by positive numberstso that the space with parameterbiscontained isometrically in the space with parameter a whenais less than b. The defining

functionE(t, z) of the space with parametertsatisfies the identities with

s(t) = 2/t.

The function is chosen so that the identities are satisfied with

p(t) = exp(t)s(t)

andr(t) = exp(t)s(t).

The functionsE(t, z) depend continuously ontand satisfy the integral equation

(A(b, z), B(b, z))I (A(a, z), B(a, z))I=z ba

(A(t, z), B(t, z))dm(t)

when a is less than bwith

m(t) =

(t) (t)(t) (t)

a nonincreasing matrixvalued function of positive t with differentiable entries such that

t(t) =p(t)/s(t)

andt(t) =r(t)/s(t)

and(t) = 0.


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The analytic weight function

W(z) = (h iz)

is treated in relation to the contiguous analytic weight functions

W(z) = (h

1

2iz)

andW+(z) = (h +

12 iz)

when h 12 is positive. Contiguity applies also to associated Hilbert spaces of entirefunctions. Assume that a spaceH(E) is contained isometrically in the weighted HardyspaceF(W) and that the space contains an entire function whenever its product with apolynomial belongs to the space. The defining function E(z) of the space is chosen tosatisfy the symmetry condition

E(z) = E(z)

since the symmetry conditionW(z) = W(z)

is satisfied. The functionE(z) is uniquely determined by the requirement that E(z) hasvalue one at the origin. Contiguous spaces H(E) contained isometrically in the weightedHardy spaceF(W) andH(E+) contained isometrically in the weighted Hardy spaceF(W+) are constructed with analogous properties.

Since multiplication byh 12 iz

is an isometric transformation of the spaceF(W) onto the spaceF(W+), the spacesH(E) andH(E+) are chosen so that the multiplication is an isometric transformation ofthe space H(E) onto the set of elements of the space H(E+) which vanish at 12 i ih. Theadjoint transformation of the space H(E+) into the spaceH(E) takes a function F(z) ofz into the function

[F(z) L(z)F(12 i ih)]/(h 12 iz)ofz with

L(z) =K+(12 i ih,z)/K+(12 i ih, 12 i ih)

the constant multiple of the reproducing kernel function for function values at 12 i ih inthe spaceH(E+) which has value one at 12 i ih. The equations

A(z) = [A+(z) L(z)A+(12 i ih)]/(h 12 iz)

and1B(z) = [B+(z) L(z)B+(12 i ih)]/(h 12 iz)

apply with= [1 L(0)A+(12 i ih)]/(h 12 )


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a nonzero real number.

A contractive transformation of the space F(W) into the space F(W) and of the spaceF(W) into the spaceF(W+) is defined by taking a function F(z) ofz into the functionF(z + 12 i) ofz . The adjoint transformation of the space F(W) into the space F(W) takesa function F(z) ofz into the function

F(z+ 12 i)/(h 12 iz)

ofz. The adjoint transformation of the space F(W+) into the space F(W) takes a functionF(z) ofz into the function

F(z+ 12 i)/(h iz)ofz .

The spacesH(E) andH(E+) are constructed so that a contractive transformation ofthe spaceH(E) into the spaceH(E) and of the spaceH(E) into the spaceH(E+) isdefined by taking a function F(z) ofz into the function F(z+ 12 i) of z and so that the

adjoint transformation of the spaceH(E) into the spaceH(E) takes a function F(z) ofz into the function

[F(z+ 12 i) L(z)F(i ih)]/(h 12 iz)ofz .

The transformation of the spaceH(E) into the spaceH(E) takes the function

[B(z)A(w) A(z)B(w)]/[(z w)]

ofz into the function

[B(z+ 12 i)A(w)

A(z+ 12 i)B(w)]/[(z+ 12 i w)]

ofz for every complex number w. The adjoint transformation of the spaceH(E) into thespaceH(E) takes the function

[B(z)A(w) A(z)B(w)]/[(z w)]

ofz into the function

B(z+ 12 i)A(w) A(z+ 12 i)B(w)(z+ 12 i w)(h 12 iz)

L(z) B(i ih)A(w) A(i ih)B(w)

(i ih w)(h 12 iz)

ofz for every complex number w.


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The identity

[B(z+ 12 i)A(w)

A(z+ 12 i)B(w)]/[(z+ 12 i w)]

= B(z)A(w+ 12 i)

A(z)B(w+ 12 i)(z+ 12 i w)(h 12 + iw)

L(w)

B(z)A(i

ih)

A(z)B(i

ih)

(z+ i ih)(h 12 + iw)

follows by properties of reproducing kernel functions for all complex numbers z and w.

The identity can be written

B(z+ 12 i)A(w)

A(z+ 12 i)B(w)=B(z)[A(w+ 12 i)

L(w)A(i ih)]/(h 12 + iw)A(z)[B(w+ 12 i) L(w)B(i ih)]/(h 12 + iw)

+L(w)[B(z)A(i

ih)

A(z)B(i

ih)]/(h

1 + iz).

Since the functions A(z),B(z), and

[B(z)A(i ih) A(z)B(i ih)]/(h 1 + iz)

ofz are linearly independent, the equations

A(z+ 12 i) =A(z)P+ B(z)R

v [B(z)A(i ih) A(z)B(i ih)]/(h 1 + iz)

andB(z+

12 i) = A(z)Q + B(z)S

+u [B(z)A(i ih) A(z)B(i ih)]/(h 1 + iz)apply for unique complex numbers u and v and for a unique matrix

P QR S

with complex entries.

Symmetry implies that u and the diagonal entries of the matrix are real and that vand the offdiagonal entries of the matrix are imaginary. The equations

A(z)S+ B(z)R= [A(z+ 12 i) L(z)A(i ih)]/(h 12 iz)

andA(z)Q + B(z)P = [B(z+

12 i) L(z)B(i ih)]/(h 12 iz)


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are satisfied withL(z) = A(z)u+ B(z)v.

Consistency of the equations implies the constraints

P S=QR

and1 = [P A(ih i) + RB(ih i)]u [QA(ih i) + SB(ih i)]v.

The recurrence relations

[A(z+ i) L(z)A(32 i ih)]/(h 12 iz)+ [B(z)A(

32 i ih) A(z)B(32 i ih)]v/(32 h iz)

= 2P[A(z)S+ B(z)R]

and

[B(z+ i) L(z)B(3

2 i ih)]/(h 1

2 iz)+ [B(z)A(

32 i ih) A(z)B(32 i ih)]u/(32 h iz)

= 2S[A(z)Q + B(z)P]

are satisfied.

The recurrence relations

[A(z+ i) L(z)A(i ih)]/(h iz)+ [B(z)A(i ih) A(z)B(i ih)]v/(1 h iz)

= 2S[A(z)P+ B(z)R]

and[B(z+ i) L(z)B(i ih)]/(h iz)

+ [B(z)A(i ih) A(z)B(i ih)]u/(1 h iz)= 2P[A(z)Q + B(z)S]

are obtained withL(z) =A(z)u + B(z)v

whereu= P u Qv+ B(ih i)uv/(h 1)

andv= Ru+ Sv+ A(ih i)uv/(h 1).

The entire function E(t, z) of z depends on a positive parameter t and satisfies thedifferential equations

t B(t, z) =zA(t, z)(t)


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and

t

A(t, z) =zB(t, z)(t)

for differentiable functions (t) and (t) of positive t. The coefficientsu(t) and v(t) andthe entries of the matrix

P(t) Q(t)R(t) S(t)

are differentiable functions oft. The derivatives(t) and(t) are negative since the spaceparametrized byb is contained in the space parametrized by a when a is less than b. Theparametrization is made so that

t(t) = 1.The constant of integration in (t) is chosen so that

t= exp((t)).

Similar constructions are made with subscripts plus and minus. The differential equa-tions

t(t) =

iQ(t)/S(t) and

t(t) =

iQ(t)/P(t)

andt(t) =iR(t)/P(t) and t(t) =iR(t)/S(t)

andu(t) = ihv(t)(t)

andv(t) = ihu(t)(t)

are obtained on differentiation.

The differential equations

Q(t)A(t,ih i)v(t) + R(t)B(t,ih i)u(t)=S(t)/S(t) + 12 t

1 = P(t)/P(t) 12 t1and

P(t)A(t,ih i)u(t) + S(t)B(t,ih i)v(t)=R(t)/R(t) + 12 t

1 = Q(t)/Q(t) 12 t1are obtained where

t P(t)S(t) = 1 =tQ(t)R(t)

and1 =P(t)A(t,ih i)u(t) S(t)B(t,ih i)v(t)

Q(t)A(t,ih

i)v(t) + R(t)B(t,ih

i)u(t).

The equations[R(t)/P(t)] =L(t,ih i) R(t)/P(t)

and[S(t)/Q(t)] =L(t,ih i) S(t)/Q(t)

are satisfied.


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2. Fourier Analysis on the Complex SkewPlane

The Hilbert spaces of entire functions constructed from the gamma function apply toFourier analysis for the complex skewplane. The complex skewplane is a vector space ofdimension four over the real numbers which contains the complex plane as a vector subspaceof dimension two. The multiplicative structure of the complex plane as a field is generalizedas the multiplicative structure of the complex skewplane as a skewfield. The conjugationof the complex plane is an automorphism which extends as an antiautomorphism of thecomplex skewplane.

An element

= t + ix +jy + kz

of the complex skewplane has four real coordinates x, y, z, andt. The conjugate elementis

=t ixjy kz.The multiplication table

ij = k,jk= i,ki= jji = k,kj = i,ki= j

defines a conjugated algebra in which every nonzero element has an inverse. An automor-phism of the skewfield is an inner automorphism which is defined by an element withconjugate as inverse. A plane is a maximal commutative subalgebra. Every plane is iso-morphic to every other plane under an automorphism of the skewfield. The complexplane is the subalgebra of elements which commute with i.

The topology of the complex skewplane is derived from the topology of the real lineas is the topology of the complex plane. Addition and multiplication are continuous as

transformations of the Cartesian product of the space with itself into the space. Thetopology of the real line is derived from Dedekind cuts. A real numbert divides the realline into two open halflines (t,) and (, t). The intersection of open halflines is anopen interval (a, b) when it is nonempty and not a halfline. A open subset of the line isa union of open intervals. The topology of the plane is the Cartesian product topology oftwo Dedekind topologies of two coordinate lines. The topology of the complex skewplaneis the Cartesian product topology of the Dedekind topologies of four coordinate lines.

The canonical measure for the complex skewplane is derived from the canonical mea-sure for the real line as is the canonical measure for the complex plane. In all cases thecanonical measure is defined on the Baire subsets of the space defined as the smallest classof sets containing the open sets and the closed sets and containing countable unions andcountable intersections of sets of the class. A measure preserving transformation of thespace onto itself is defined by taking into + for every element of the space. Thiscondition determines the canonical measure within a constant factor.

The canonical measure for the real line is Lebesgue measure, which assigns measure oneto the interval (0, 1). The canonical measure for the complex plane is the Cartesian productmeasure of the canonical measure of two coordinate lines, which assigns measure to the


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unit disk zz < 1. The canonical measure for the complex skewplane is the Cartesianproduct measure of the canonical measures of four coordinate lines, which assigns measure12

2 to the unit disk


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for every rational number. The conjugate character is primitive modulo when thegiven character is primitive modulo .

The integral elements of the complex plane are according to Gauss the elements whosecoordinates are integral elements of the line. An Euclidean algorithm applies to the ringof integral elements of the complex plane. If is an integral element and ifis a nonzerointegral element, then an integral element exists which satisfies the inequality

| | < ||.

An ideal of the ring of integral elements which contains a nonzero element contains anelement which minimizes the positive integer . Every element of the ideal is aproduct

=

with an element of the ring.

An application of the Euclidean algorithm for the plane is the representation of a positiveinteger as the sum of two squares

a2 + b2 = (a + ib)(a + ib)

for integersa and b. The representation problem reduces to one for prime numbers by themultiplicative structure of the complex plane. The even prime

2 = 1 + 1

is a sum of two squares. A prime which is congruent to three modulo four is not a sum oftwo squares since the representation is not possible modulo four. A prime

p= a2 + b2

is a sum of two squares of integers aand b if it is congruent to one modulo four.

The integral elements of the complex skewplane are according to Hurwitz the elementswhose coordinates are either all integral elements of the line or all halfintegral elementsof the line. An Euclidean algorithm applies to the ring of integral elements of the complexskewplane. If is an integral element and if is a nonzero integral element, then anintegral element exists which satisfies the inequality

|

|

Proof Riemann

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