Γ(1 s) Scattering, determinants, hyperfunctions in relation to Γ(−s) Jean-Franc¸ois Burnol 8 0 0 Abstract 2 n The method of realizing certain self-reciprocal transforms as (absolute) scattering, previ- a ously presented in summarized form in the case of the Fourier cosine and sine transforms, is J here applied to the self-reciprocal transform f(y) 7→ H(f)(x) = R0∞J0(2√xy)f(y)dy, which 2 is isometrically equivalent to the Hankel transform of order zero and is related to the func- tional equations of the Dedekindzeta functions of imaginary quadratic fields. This also allows ] to re-prove and to extend theorems of de Branges and V. Rovnyak regarding square inte- T grable functions which are self-or-skew reciprocal under the Hankel transform of order zero. N Related integral formulae involving various Bessel functions are all established internally to h. the method. Fredholm determinants of the kernel J0(2√xy) restricted to finite intervals (0,a) givethe coefficients of first and second order differential equations whose associated scattering t a is (isometrically) the self-reciprocal transform , closely related to the function Γ(1−s). Re- m H Γ(s) markable distributions involved in this analysis are seen to have most natural expressions as [ (differenceof)boundaryvalues(i.e. hyperfunctions.) Thepresentwork iscompletely indepen- dent from the previous study by the author on the same transform , which centered around 2 H theKlein-Gordonequationandrelativisticcausality. Inanappendix,wemakeasimple-minded v observationregardingtheresolventoftheDirichletkernelasaHilbertspacereproducingkernel. 5 2 4 keywords: Hankeltransform;Scattering; Fredholmdeterminants. 2 0 Universit´eLille1 6 UFRdeMath´ematiques 0 Cit´escientifiqueM2 / F-59655Villeneuved’Ascq h France t [email protected] a m : v February19,2006. i January2,2008.: therewasaniin(148c)andotherequationsleadingtoTheorem15whichshouldnot X have been there. This only affected equation (156e). This is all folks. I could possibly have proposed r other improvements if only the referee who kept my paper hostage for most of 2006 and 2007 had a actuallyreadit. Notreadingitdidnotpreventfromcommentinguponit,unfortunatelythesubstance ofthoseinspiredfivelinesishardtotransferbeneficiallytomywidereadership. 1 Contents 1 Introduction (the idea of co-Poisson) 2 2 Hardy spaces and the de Branges-Rovnyak isometric expansion 8 3 Tempered distributions and their and Mellin transforms 11 H 4 A group of distributions and related integral formulas 16 5 Orthogonal projections and Hilbert space evaluators 18 6 Fredholm determinants, the first order differential system, and scattering 28 7 The K-Bessel function in the theory of the transform 39 H 8 The reproducing kernel and differential equations for the extended spaces 44 9 Hyperfunctions in the study of the transform 54 H 10 Appendix: a remark on the resolvent of the Dirichlet kernel 60 1 Introduction (the idea of co-Poisson) Weexplaintheunderlyingframeworkandthegeneralcontoursofthiswork. Throughoutthepaper, we have tried to formulate the theorems in such a form that one can, for most of them, read their statements without having studied the preceeding material in its entirety, so a sufficiently clear idea of the results and methods is easily accessible. Setting up here all notations and necessary preliminaries for stating the results would have taken up too much space. The Riemann zeta function ζ(s) = 1 + 1 + 1 +... is a meromorphic function in the entire 1s 2s 3s complex plane with a simple pole at s = 1, residue 1. Its functional equation is usually written in one of the following two forms: s s 1−s 1 s π−2Γ( )ζ(s)=π− 2 Γ( − )ζ(1 s) (1a) 2 2 − Γ(1 s) ζ(s)=χ0(s)ζ(1−s) χ0(s)=πs−21 Γ(−2s) (1b) 2 Theformerisrelatedtotheexpressionofπ−2sΓ(2s)ζ(s)asaleftMellintransform1 andtotheJacobi identity: π−s2Γ(s)ζ(s)= 1 ∞(θ(t) 1)ts2−1dt ( (s)>1) (2a) 2 2 − ℜ Z0 = 1 ∞(θ(t) 1 1 )ts2−1dt (0< (s)<1) (2b) 2 − − √t ℜ Z0 1 1 θ(t)=1+2 qn2 q =e πt θ(t)= θ( ) (2c) − √t t n 1 X≥ 1intheleftMellintransformweuses 1,intherightMellintransformweuse s. − − 2 The latter form of the functional equation is related to the expression of ζ(s) as the right Mellin transform of a tempered distribution with support in [0,+ ), which is self-reciprocal under the ∞ Fourier cosine transform:2 ζ(s)= ∞( δ (x) 1)x sdx (3a) m − − Z0 m 1 X≥ ∞ 2cos(2πxy)( δ (y) 1)dy = δ (x) 1 (x>0) (3b) n m − − Z0 n 1 m 1 X≥ X≥ This last identity may be written in the more familiar form: e2πixy δ (y)dy = δ (x) (4) n m ZR n Z m Z X∈ X∈ which expresses the invariance of the “Dirac comb” distribution δ (x) under the Fourier m Z m ∈ transform. As a linear functional on Schwartz functions φ , the invariance of δ (x) under P m Z m ∈ Fourier is expressed as the Poisson summation formula: P φ(n)= φ(m) φ(y)= e2πixyφ(x)dx (5) n Z m Z ZR X∈ X∈ e e The Jacobi identity is the special instance with φ(x) = exp( πtx2), and conversely the validity of − (5) for Schwartz functions (and more) may be seen as a corollaryto the Jacobi identity. Theidea of co-Poisson [4]leadstoanother formulationofthefunctionalequationasanidentity involving functions. The co-Poisson identity ((10) below) appeared in the work of Duffin and Weinberger[13]. Inoneoftheapproachestothisidentity,westartwithafunctiong onthepositive half-line such that both 0∞g(t)dt and 0∞g(t)t−1dt are absolutely convergent. Then we consider the averaged distributionRg∗D(x) = 0∞R g(t)D(xt)dtt where D(x) = n≥1δn(x)−1x>0(x). This gives (for x>0): R P ∞ g(x/n) ∞ g(1/t) g D(x)= dt (6) ∗ n − t n=1 Z0 X If g is smooth with support in [a,A], 0<a<A< , then the co-Poissonsum g D has Schwartz ∞ ∗ decrease at + (easy from applying the Poisson formula to g(1/t); cf. [8, 4.29] for a general ∞ t statement). The right Mellin transform g[D(s) is related to the right Mellin transform g(s) of g ∗ via the identity: b g[D(s)= ∞(g D)(x)x−sdx=ζ(s) ∞g(x)x−sdx=ζ(s)g(s) (7) ∗ ∗ Z0 Z0 This is because the right Mellin transform of a multiplicative convolution isbthe product of the rightMellin transforms. The necessarycalculus oftempereddistributions neededforthis andother statements in this paragraphis detailed in [8]. The functional equation in the form of (1b) gives:3 g(1/t) g[D(s)=χ (s)ζ(1 s)g(s)=χ (s)I(\g) D(1 s) I(g)(t)= (8) 0 0 ∗ − ∗ − t 2ofcourse,δm(x)=δ(x−m). b 3oneobserves thatI(g)(s)=g(1 s). d b − 3 One may reinterpret this in a manner involving the cosine transform acting on L2(0,+ ;dx). C ∞ The Mellin transform of a function f(x) in L2(0, ;dx) is a function f(s) on (s) = 1 which is ∞ ℜ 2 nothing else than the Plancherel Fourier transform of e12uf(eu): f(12 +ibγ)= 0∞f(x)x−21−iγdx = −∞∞f(eu)eu2e−iγudu, 0∞|f(x)|2dx = −∞∞|f(eu)eu2|2du = 21πbℜ(s)=12 |f(s)R|2|ds|. The unitary \ RoperatorCI isscaleinvaRrianthenceitisdiRagonalizedbytheMellintRransformb: CI(f)(s)=χ0(s)f(s), (f)(s)=χ (s)f(1 s),whereχ (s)isobtainedforexampleusingf(x)=e πx2 andcoincideswith 0 0 − C − b the chi-function defined in (1b). It has modulus 1 on the critical line as is unitary. So (8) says d b C that the co-Poisson intertwining identity holds: (g D)=I(g) D (9) C ∗ ∗ The co-Poissonintertwining (9) or explicitely: ∞ ∞ g(x/m) ∞ g(1/t) ∞ g(n/y) ∞ 2cos(2πxy) dt dx= g(t)dt (y >0) (10) Z0 m=1 m −Z0 t ! n=1 y −Z0 X X is, when g is smooth with support in [a,A], 0 < a < A < , an identity of (even) Schwartz ∞ functions. If g is only supposed to be such that ∞ g(t)(1 + 1)dt < then the co-Poisson 0 | | t ∞ intertwining (g D) = I(g) D holds as an identRity of distributions (either considered even or C ∗ ∗ with support in [0, )). Sufficient conditions for pointwise validity have been established [8]. The ∞ general statement of the intertwining is (g E) = I(g) (E) where E is an arbitrary tempered C ∗ ∗C distributionwithsupporton[0, )(seefootnote4)anditisprovendirectly. Theco-Poissonidentity ∞ (10)isanothermanner,notidenticalwiththePoissonsummationformula,toexpresstheinvariance of D under the cosine transform, or the invariance of the Dirac comb under the Fourier transform. If the integrable function g has its support in [a,A], 0 <a < A< , then g D is constant in ∞ ∗ (0,a) and its cosine transform (thanks to the co-Poisson intertwining) is constant in (0,A 1). Up − toarescalingwemaytakeA=a 1,andthena<1(ifanonzeroexampleiswanted.) Appropriate − modifications allow to construct non zero even Schwartz functions constant in ( a,a) and with − Fourier transform again constant in ( a,a) where a>0 is arbitrary [8]. − Schwartzfunctions aresquare-integrablesohere we havemade contactwiththe investigationof de Branges [1], V Rovnyak [28] and J. and V. Rovnyak [29, 30] of square integrable functions on (0, ) vanishing on (0,a) and with Hankel transform of order ν vanishing on (0,a). For ν = 1 ∞ −2 the Hankel transform of order ν is f(y) 2 ∞cos(xy)f(y)dy and up to a scale change this 7→ π 0 is the cosine transform considered above. Tqhe co-Poisson idea allows to attach the zeta function R to, among the spaces defined by de Branges [1], the spaces associated with the cosine transform: it has allowed the definition of some novel Hilbert spaces [3] of entire functions in relation with the Riemann zeta function and Dirichlet L-functions (the co-Poisson idea is in [4] on the adeles of an arbitrary algebraic number field K; then, the study of the related Hilbert spaces was begun for K =Q. Further results were obtained in [7].) 4both sides in fact depend only on E(x)+E( x) as a distribution on the line, which may be identically 0, and − thishappensexactlywhen E isalinearcombinationofoddderivativesofthedeltafunction. 4 Thestudyofthefunctionχ0(s)=πs−12ΓΓ((1−s2s)),ofunitmodulusonthecriticalline,isinteresting. 2 Weproposedtorealizetheχ functionasa“scatteringmatrix”. Thisisindeedpossibleandhasbeen 0 achieved in [6]. The distributions, functions, and differential equations involved are all related to, orexpressedby,the Fredholmdeterminantsofthe finite cosinetransform,whichinturnarerelated to the Fredholm determinants of the finite Dirichlet kernels sin(t(x−y)) on [ 1,1]. The study of the π(x y) − − Dirichlet kernels is a topic with a vast literature. A minor remark will be made in an appendix. We mentioned the Riemann zeta function and how it relates to χ0(s)=πs−12ΓΓ(1(−s2s)) and to the 2 cosine transform. Let us now briefly consider the Dedekind zeta function of the Gaussian number field Q(i) and how it relates to χ(s)= Γ(1−s) and to the transform. The transform is Γ(s) H H (g)(y)= ∞J (2√xy)g(x)dx J (2√xy)= ∞ ( 1)nxnyn (11) H 0 0 − n!2 Z0 n=0 X Up to the unitary transformation g(x) = (2x)−14f(√2x), (g)(y) = (2y)−14k(√2y), it becomes H the Hankel transform of order zero k(y) = 0∞√xyJ0(xy)f(x)dx. It is a self-reciprocal, unitary, scale reversing operator ( (g(λx))(y) = 1 (g)(y)). We shall also extend its action to tempered H λHR λ distributions on R with support in [0,+ ). At the level of right Mellin transforms of elements of ∞ L2(0, ;dx) it acts as: ∞ Γ(1 s) 1 [ (g)(s)=χ(s)g(1 s) χ(s)= − (s)= (12) H − Γ(s) ℜ 2 Ithase x1 (x)asoneamongitssbelf-reciprocalfunctions,asisverifieddirectlybyseriesexpansion − x 0 0∞J0(2√xy≥)e−ydy = ∞n=0 (−n1!2)nxn 0∞yne−ydy =e−x. The identity R P ∞J0R(2√t)t−sdt=χ(s)= Γ(1−s) (13) Γ(s) Z0 is equivalent to a special case of well-known formulas of Weber, Sonine and Schafheitlin [33, 13.24.(1)]. Here we have an absolutely convergent integral for 3 < (s) < 1 and in that range 4 ℜ the identity may be proven as in: e−x = 0∞J0(2√xy)e−ydy = 0∞J0(2√y)x1e−xy dy, Γ(1−s) = 0∞J0(2√y)( 0∞x−s−1e−yx dx)dy = Γ(s)R0∞J0(2√y)y−sdy. ThRe integral is semi-convergent for (s) > 1, and of course (13) still holds. In particular on the critical line and writing t = eu, ℜR 4 R R s= 12 +iγ, we obtain the identities of tempered distributions Re12uJ0(2e21u)e−iγudu=χ(21 +iγ), e12uJ0(2e21u)= 21π Rχ(12 +iγ)e+iγudu. R We have ζ (sR) = 1 1 = 1 + 1 + 1 + 2 + 1 + = cn and it Q(i) 4 (n,m)=(0,0) (n2+m2)s 1s 2s 4s 5s 8s ··· n 1 ns 6 ≥ is a meromorphic function in the entire complex plane with a simple pole at s = 1, residue π. Its P P 4 functional equation assumes at least two convenient well-known forms: (√4)s(2π) sΓ(s)ζ (s)=(√4)1 s(2π) (1 s)Γ(1 s)ζ (1 s) (14a) − Q(i) − − − Q(i) − − 1 1 Γ(1 s) (π)sζQ(i)(s)=χ(s)(π)1−sζQ(i)(1−s) χ(s)= Γ(−s) (14b) 5 The former is related to the expression of π sΓ(s)ζ (s) as a left Mellin transform: − Q(i) π sΓ(s)ζ (s)= 1 ∞(θ(t)2 1)ts 1dt ( (s)>1) (15a) − Q(i) 4 − − ℜ Z0 = 1 ∞(θ(t)2 1 1)ts 1dt (0< (s)<1) (15b) − 4 − − t ℜ Z0 1 1 θ(t)2 = θ( )2 (15c) t t The latter form of the functional equation is related to the expression of (1)sζ (s) as the right π Q(i) Mellintransformofatempereddistributionwhichissupportedin[0, )andwhichisself-reciprocal ∞ under the -transform: H (1)sζ (s)= ∞( c δ (x) 1)x sdx (16a) π Q(i) m πm − 4 − Z0 m 1 X≥ ∞ 1 1 J (2√xy)( c δ (y) )dy = c δ (x) 1 (x)=E(x) (x>0) (16b) 0 n πn m πm x>0 − 4 − 4 Z0 n 1 m 1 X≥ X≥ The invariance of E under the -transform is equivalent to the validity of the functional equation H of (1)sζ (s) and it having a pole with residue 1 at s = 1. The co-Poisson intertwining becomes π Q(i) 4 the assertion: ∞ ∞ g(x/πm) 1 ∞ 1 dt ∞ g(πn/y) 1 ∞ y >0 = J (2√xy) c g( ) dx= c g(t)dt 0 m n ⇒ Z0 m=1 πm − 4Z0 t t ! n=1 y −4Z0 X X (17) If g is smooth with support in [b,B], 0 < b < B < , then we have on the right hand side ∞ a function of Schwartz decrease at + (compare to Theorem 3), and its -transform is also of ∞ H Schwartz decrease at + . The former is constant for 0< y < πB 1 and the latter is constant for − ∞ 0<x<πb. The supremum of the values obtainable for the product of the lengths of the intervals of constancy is π2. But, as for the cosine and sine transforms, there does exist smooth functions which are constant on a given (0,a) for arbitrary a > 0 with an transform again constant on H (0,a) and have Schwartz decrease at + (the two constants being arbitrarily prescribed.) ∞ De Branges and V. Rovnyak have obtained [1, 28] rather complete results in the study of the Hankel transform of order zero f(x) 7→ g(y) = 0∞√xyJ0(xy)f(x)dx from the point of view of understanding the support property of being zeroRand with transformagainzero in a giveninterval (0,b). They obtained an isometric expansion (Theorem 1 of section 2) and also the detailed de- scription of the related spaces of entire functions ([1]). The more complicated case of the Hankel transforms of non-zero integer orders was treated by J. and V. Rovnyak [29, 30]. These, rather complete, results are an indication that the Hankel transform of order zero or of integer order is easier to understand than the cosine or sine transforms, and that doing so thoroughly could be useful to better understand how to try to understand the cosine and sine transforms. The kernel J (2√uv) of the -transform satisfies the Klein-Gordon equation in the variables 0 H 6 x=v u, t=v+u: − ∂2 ∂2 ∂2 ( +1)J (2√uv)=((cid:3)+1)J (2√uv)=( +1)J ( t2 x2)=0 (18) ∂u∂v 0 0 ∂t2 − ∂x2 0 − p Itisanoteworthyfactthatthesupportcondition,initiallyconsideredbydeBrangesandV.Rovnyak, andwhich, nowadays,is alsoseen to be in relationwith the co-Poissonidentities, has turnedout to be related to the relativistic causality governing the propagation of solutions to the Klein-Gordon equation. This has been established in [9] where we obtained as an application of this idea the isometric expansion of [1, 28] in a novel manner. It was furthermore proven in [9] that the H transform is indeed an (absolute) scattering, in fact the scattering from the past boundary to the future boundary of the Rindler wedge 0 < t < x for solutions of a first order, two-component | | (“Dirac”), form of the KG equation. In the present paper, which is completely independent from [9], we shall again study the - H transform and show in particular how to recover in yet a different way the earlier results of [1, 28] and also we shall extend them. This will be based on the methods from [5, 6], and uses the techniques motivated by the study of the co-Poisson idea [8]. Our exposition will thus give a fully detailed account of the material available in summarized form in [5, 6]. Then we proceed with a development of these methods to provide the elucidation of the (two dimensions bigger) spaces of functions constant in (0,a) and with -transforms constant in (0,a). H Theuseoftempereddistributionsisanimportantpointofourapproach5; alsoonemayenvision the co-Poisson idea as asking not to completely identify a distribution with the linear functional it “is”. In this regard it is of note that the distributions which arise following the method of [5] are seen in the present case of the study of the -transform to have a very natural formulation as H differences of boundary values of analytic functions, that is, as hyperfunctions [23]. We do not use the theory of hyperfunctions as such, but could not see how not to mention that this is what these distributions seem to be in a natural manner. The paper contains no number theory. And, the reader will need no prior knowledge of [2]; some familiarity with the m-function of Hermann Weyl [10, 21, 26] is necessary at one stage of the discussion (there is much common ground, in fact, between the properties of the m-function and the axioms of [2]). The reproducing kernel in any space with the axioms of [2] has a specific appearance (equation (109) below) which has been used as a guide to what we should be looking for. The validity of the formula is re-proven in the specific instance considered here6. Regarding the differential equations governing the deformation, with respect to the parameter a >0 7, of the 5at the bottom of page 456 of [1] the formulas given for A(a,z) and B(a,z) as completed Mellintransforms are lacking terms which would correspond to Dirac distributions; possibly related to this, the isometric expansion as presentedinTheoremIIof[1]islackingcorrespondingterms. Theexactisometricexpansionappearsin[28]andthe exactformulasforA(a,z)andB(a,z)ascompletedMellintransformsappear,inanequivalentform,in[30,eq.(37)]. 6thecriticallinehereplays theroˆleofthe realaxisin[2],sis 1 iz andthe useof thevariablesismostuseful 2 − in distinguishing the right Mellin transforms which need to be completed by a Gamma factor from the left Mellin transformsof“theta”-likefunctions. 7theaherecorrespondsto 1a2 in[1]. 2 7 Hilbert spaces,wedepartfromthe generalformalismof[2]andobtainthem ina canonicalform,as defined in [21, 3]. Interestingly this is related to the fact that the A and B functions (connected § to the reproducing kernel, equation (109)) which are obtained by the method of [5] turn out not to be normalized according to the rule in general use in [2]. Each rule of normalization has its own advantages; here the equations are obtained in the Schro¨dinger and Dirac forms familiar from the spectral theory of linear second order differential equations [10, 21, 26]. This allows to make reference to the well-knownWeyl-Stone-Titchmarsh-Kodairatheory [10,21, 26], and to understand as a scattering. Regarding spaces with the axioms of [2], the articles of Dym [14] and Remling H [27]willbeusefultothereaderinterestedinsecondorderlineardifferentialequations. Andwerefer the reader with number theoretical interests to the recent papers of Lagarias [18, 19]. The author has been confronted with a dilemma: a substantial portion of the paper (most of chapters 5, 6, 8) has a general validity for operators having a kernel of the multiplicative type k(xy) possessing certain properties in common with the cosine, sine or transforms. But on the H other hand the (essentially) unique example where all quantities arising may be computed is the transform (and transforms derived from it, or closely related to it, as the Hankel transforms of H integer orders). We have tried to give proofs whose generality is obvious, but we also made full use of distributions, as this allows to give to the quantities arising very natural expressions. Also we never hesitate using arguments of analyticity although for some topics (for example, some aspects involvingcertainintegralequations andFredholm determinants)this is certainly not reallyneeded. 2 Hardy spaces and the de Branges-Rovnyak isometric ex- pansion Let us state the isometric expansionof [1, 28] regardingthe square integrable Hankel transforms of order zero. We reformulate the theorem to express it with the transform (11) rather than the H Hankel transform of order zero. Theorem 1 ([1], [28]). : Let k L2(0, ;dx). The functions f and g , defined as the following 1 1 ∈ ∞ integrals: ∞ f (y)= J (2 y(x y))k(x)dx, (19a) 1 0 − Zy p ∞ y g (y)=k(y) J (2 y(x y))k(x)dx, (19b) 1 1 − x y − Zy r − p exist in L2 in the sense of mean-square convergence, and they verify: ∞ f (y)2+ g (y)2dy = ∞ k(x)2dx. (19c) 1 1 | | | | | | Z0 Z0 The function k is given in terms of the pair (f ,g ) as: 1 1 x x y k(x)=g (x)+ J (2 y(x y))f (y)dy J (2 y(x y))g (y)dy (19d) 1 0 1 1 1 − − x y − Z0 Z0 r − p p 8 Theassignmentk (f ,g )isaunitaryequivalenceofL2(0, ;dx)withL2(0, ;dy) L2(0, ;dy) 1 1 7→ ∞ ∞ ⊕ ∞ such that the -transform acts as (f ,g ) (g ,f ). Furthermore k and (k) both identically 1 1 1 1 H 7→ H vanish in (0,a) if and only if f and g both identically vanish in (0,a). 1 1 Let us mention the following (which follows from the proof we have given of Thm. 1 in [9]): if f , f , g , g are in L2 then k, k and (k) are in L2. Conversely if k, k and (k) are in L2 then 1 1′ 1 1′ ′ H ′ ′ H ′ the integrals defining f (y) and g (y) are convergent for each y >0 as improper Riemann integrals, 1 1 and f and g are in L2. 1′ 1′ It will prove convenient to work with (f(x),g(x))= 1(g (x)+f (x),g (x) f (x)): 2 1 2 1 2 1 2 − 1 2 1 y 1 ∞ y f(y)= k( )+ J ( y(2x y)) J ( y(2x y)) k(x)dx (20a) 0 1 2 2 2 − − 2x y − Zy/2(cid:18) r − (cid:19) p p 1 y 1 ∞ y g(y)= k( ) J ( y(2x y))+ J ( y(2x y)) k(x)dx (20b) 0 1 2 2 − 2 − 2x y − Zy/2(cid:18) r − (cid:19) p p 1 2x y k(x)=f(2x)+ J ( y(2x y)) J ( y(2x y)) f(y)dy 0 1 2 − − 2x y − Z0 (cid:18) r − (cid:19) 1 2x p y p +g(2x) J ( y(2x y))+ J ( y(2x y)) g(y)dy (20c) 0 1 − 2 − 2x y − Z0 (cid:18) r − (cid:19) p p ∞ k(x)2dx= ∞ f(y)2+ g(y)2dy (20d) | | | | | | Z0 Z0 The transform on k acts as (f,g) (f, g). The pair (k, (k)) identically vanishes on (0,a) H 7→ − H if and only if the pair (f,g) identically vanishes on (0,2a). The structure of the formulas is more apparent after observing (x,y >0): ∂ 1 1 y J ( y(2x y))1 (y) =δ (y) J ( y(2x y))1 (y) (21) 0 0<y<2x 2x 1 0<y<2x ∂x 2 − − 2 2x y − (cid:16) p (cid:17) r − p In this section I shall prove the existence of an isometric expansion k (f,g) having the stated ↔ support properties and relation to the -transform; that this construction does give the equations H (20a), (20b), (20c), will only be established in the last section (9) of the paper. The method followedinthissectioncoincidespartlywiththeoneofV.Rovnyak[28];wetrytoproducethemost direct arguments, using the commonly known facts on Hardy spaces. The reader only interested in Theorem1 isinvitedafter havingreadthe presentsectiontothen jump directlyto section9 forthe conclusion of the proof. To a function k L2(0, ;dx) we associate the analytic function ∈ ∞ k(λ)= ∞eiλxk(x)dx ( (λ)>0) (22) ℑ Z0 with boundary values for λ eR again written k(λ), which defines an element of L2(R,dλ), the ∈ 2π assignment k k being unitary from L2(0, ;dx) onto H2( (λ > 0),dλ). Next we have the 7→ ∞e ℑ 2π conformal equivalence and its associated unitary map from H2( (λ>0),dλ) to H2(w <1, dθ): e ℑ 2π | | 2π λ i 1 λ+i w= − K(w)= k(λ) (23) λ+i √2 i e 9 It is well known that this indeed unitarily identifies the two Hardy spaces. With k (x) = e x, 0 − k0(λ) = λ+ii, K0(w) = √12, and kk0k2 = 0∞e−2xdx = 12 = kK0k2. The functions kn(λ) = (λ i)n i correspond to K (w) = 1 wn. To obtain explicitely the orthogonal basis (k ) , we eλ−+i λ+i n √2 R nen≥0 first observe that w =1 2 i , so as a unitary operator it acts as: − λ+i x x w k(x)=k(x) 2 e (x y)k(y)dy =k(x) e x2 eyk(y)dy (24) − − − · − − Z0 Z0 Writing k (x)=P (x)e x we thus obtain P (x)=P (x) 2 xP (y)dy: n n − n+1 n − 0 n x n n n R( 2x)j P (x)= 1 2 1= − (25) n − · j j! (cid:18) Z0 (cid:19) j=0(cid:18) (cid:19) X So as is well-known P (x) = L(0)(2x) (in the notation of [31, 5]) where the Laguerre polynomials n n § L(0)(x) are an orthonormal system for the weight e xdx on (0, ). n − ∞ One of the most common manner to be led to the -transform is to define it from the two- H dimensional Fourier transform as: (f)(1r2)= 1 ei(x1y1+x2y2)f(y12+y22)dy dy = ∞ 2πeirscosθdθ f(1s2)sds 1 2 H 2 2π 2 2π 2 ZZ Z0 (cid:18)Z0 (cid:19) (26) (f)(1r2)= ∞J (rs)f(1s2)sds r2 =x2+x2,s2 =y2+y2 H 2 0 2 1 2 1 2 Z0 which proves its unitarity, self-adjointness, and self-reciprocal character and the fact that it has e x has self-reciprocal function. The direct verification of (k ) = k is immediate: (k )(x) = − 0 0 0 H H 0∞J0(2√xy)e−ydy = ∞n=0 (−n1!2)nxn 0∞yne−ydy =e−x. Then, H(e−tx)=t−1e−xt for each t>0. RSo 0∞e−txH(k)(x)dxP=t−1 0∞e−1txRk(x)dx hence: R R i 1 k L2(0, ;dx) ](k)(λ)= k(− ) (27) ∀ ∈ ∞ H λ λ With the notation (K) for the function in H2(w < 1) correesponding to (k), we obtain from H | | H (23), (27), an extremely simple result:8 (K)(w)=K( w) (28) H − This obviously leads us to associate to K(w)= ∞n=0cnwn the functions: P ∞ F(w):= c wn (29a) 2n n=0 X ∞ G(w):= c wn (29b) 2n+1 n=0 X K(w)=F(w2)+wG(w2) (29c) and to k the functions f and g in L2(0,+ ;dx) corresponding to F and G. Certainly, k 2 = ∞ k k f 2+ g 2, and the assignmentof (f,g) to k is an isometric identification. Furthermore,certainly k k k k 8wealsotakenoteoftheoperatoridentity w= w . H· − ·H 10