# The Range of the Spectral Projection Associated with the Dunkl Laplacian

Authors: Salem Ben Said; Hatem Mejjaoli
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Fors∈ℝ, denote by Pksf the “projection” of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2. The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.


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## Body

## 1. Introduction

Analysis of the Dunkl Laplacian operatorΔk on ℝd commenced in the early 90’s, inspired by numerous results in the Euclidean setting, as well as some extensions of this to flat symmetric spaces. Here, the parameter k comes from Dunkl’s theory of differential-difference operators [1]. In recent years, there have been increasing interests in the study of problems involving the Dunkl Laplacian and have received a lot of attention, see for instance [1–15]. The purpose of this paper is to study a family of eigenfunctions for the Dunkl Laplacian derived through the use of the inversion formula for the Dunkl transform. Our main result may be interpreted as a contribution to the spectral theory of the Dunkl Laplacian.To state our main result, we need to introduce some notation. Writing the inversion formula for the Dunkl transform in polar coordinates, we obtain(1)fx=∫0∞s2λk+1fksxds,f∈Dℝd,where fks are “projections” of f into the eigenspaces of Δk corresponding to the eigenvalue −s2 (see (35)). We may also write the projection operators f↦fks as Dunkl-convolution with a normalized Bessel function of the first kind (see (43)). In this paper, we discuss on Dℝd how properties of f are related to properties of the eigenfunctions fks. Essentially, we prove a theorem characterizing fks for f∈Dℝd with suppf⊂BO,R¯, involving analytic continuation to s∈ℂ and growth estimates of type, for all N∈ℕ and for all multi-index m∈ℕd,(2)∣∂xmfksx∣≤Ck,m,N∥x∥1+∣s∣−NeR+∥x∥Ims,x∈ℝd,where Ck,m,N is a positive continuous increasing function on ℝ+ (see Theorem 9). Several contributions have been dedicated to this subject, see for instance [16–22].As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transformFk given in [23]. More precisely, we characterize the set of functions φs,η defined on ℝ×Sd−1 for which there exists a compactly supported smooth function f with support in BO,R¯ so that φs,η=Fkfsη (see Theorem 10).


## 2. Background

Forx,y∈ℝd, we let x,y denote the usual Euclidean inner product of ℝd and ∥x∥≔x,x the Euclidean norm. Let Sd−1 be the unit sphere in ℝd. We denote by dσ as the Lebesgue surface measure on Sd−1.For a nonzero vectorα∈ℝd define the reflection rα by
(3)rαx≔x−2α,x/∥α∥2α,x∈ℝd.A root system is a finite setR of nonzero vectors in ℝd such that α,β∈R implies rαβ∈R. If, in addition, α,β∈R and α=cβ for some scalar c implies c=±1, then R is called reduced. Henceforth, we will assume that R is a reduced root system. Fix a set of positive roots R+, so that R=R+∪−R+.The finite reflection groupG generated by the root system R is the subgroup of the orthogonal group Od generated by the reflections rα:α∈R+.For a given root systemR, a multiplicity function k:R⟶ℝ+;α↦kα is a nonnegative G-invariant function.Given a reduced root systemR on ℝd and a multiplicity function k=kαα∈R, we define the weight function ϑk by
(4)ϑkx≔∏α∈R+α,x2kα,x∈ℝd.Then,ϑk is a positively homogeneous G-invariant function of degree 2k, where
(5)k≔∑α∈R+kα.The main ingredient of the Dunkl theory is a family of commuting first-order differential-difference operators,Tξk (called the Dunkl operators [1]), defined by
(6)Tξkfx≔∂ξfx+∑α∈R+kαfx−frαx/α,xα,ξ,ξ∈ℝd,where ∂ξ is the ordinary partial derivative with respect to ξ. The Dunkl operators are akin to the partial derivatives and they can be used to define the Dunkl Laplacian Δk, which plays the role similar to that of the ordinary Laplacian Δ,
(7)Δkfx≔∑i=1dTξik2fx=Δfx+2∑α∈R+kα∇fx,α/α,x−∑α∈R+kαfx−frαx/α,x2∥α∥2,where ξ1,⋯,ξd is an orthonormal basis of ℝd,·,·. The above explicit expression of Δk has been proved in [24].For arbitrary finite reflection groupG, and for any nonnegative multiplicity function k, there is a unique linear operator Vk on the space of algebraic polynomials on ℝd that intertwines between the Dunkl operators and the partial derivatives,
(8)TξkVk=Vk∂ξ,∀ξ∈ℝd,Vk1=1.It has been proved in [25] that Vk has a Laplace type representation which allows to extend Vk to larger function spaces:
(9)Vkfx=∫ℝdfξdμxkξ,with a unique probability measure μxk∈M1ℝd. In fact, Vk induces a homeomorphism of Cℝd and also that of C∞ℝd; see [23, 26].Forx,y∈ℝd, define
(10)Ekx,y≔Vke·,yx.For fixedy, the function Ek·,y is the unique real-analytic solution of Tξkfx=y,ξfx with f0=1 (see [13, 27]). Further, the (Dunkl) kernel Ek has a unique holomorphic extension to ℂd×ℂd and satisfies the following properties:Fact 1 (see, for instance, [11, 25]).
(1)
For allz,w∈ℂd and λ∈ℂ, we have Ekz,w=Ekw,z and Ekλz,w=Ekz,λw.(2)
For every multi-indexm∈ℕd, we have(11)∣∂zmEkx,z∣≤∥x∥∣m∣e∥x∥∥Rez∥,where ∣m∣=m1+⋯+md. In particular, ∣Ekx,iy∣≤1 for all x,y∈ℝd.
Forf∈L1ℝd,ϑkxdx, the Dunkl transform is defined by
(12)Fkfξ≔ck−1∫ℝdfxEkx,−iξϑkxdx,ξ∈ℝd,where ck is the constant
(13)ck≔∫ℝde−∥x∥2/2ϑkxdx.
The Dunkl transform was introduced in [28] where the L2-isometry (or the Plancherel theorem) was proved, while the main results of the L1-theory were established in [11]. In particular, it has been proved that if f and Fkf are in L1ℝd,ϑkxdx, then for almost every x∈ℝd,(14)fx=ck−1∫ℝdFkfξEkξ,ixϑkξdξ,ξ∈ℝd.
It is worth mentioning that the Dunkl transform is a homeomorphism of the Schwartz spaceSℝd. Further, according to ([29], Proposition 5.7.8), for f∈L1ℝd,ϑkxdx such that fx=f0∥x∥ with f0:ℝ+⟶ℂ, we have
(15)Fkfξ=Hλkf0∥ξ∥,where
(16)λk≔k+d−2/2,and Hα is the Hankel transform of index α on L1ℝ+,r2α+1dr, given by
(17)Hαgs=1/2αΓα+1∫0∞grjαrsr2α+1dr.
Here,jα is the normalized Bessel function of the first kind defined by
(18)jαz≔Γα+1z/2−αJαz,whereJαz=z/2α∑n=0∞−1nz/22n/n!Γα+n+1.
Lety∈ℝd be given. For f∈Sℝd, the generalized translation operator is defined by
(19)τyfx≔ck−1∫ℝdFkfξEkix,ξEkiy,ξϑkξdξ,x∈ℝd.Fact 2 (see [26]).
The translation operator has the following properties:(1)
For allx,y∈ℝd,τyfx=τxfy.(2)
For fixedy∈ℝd,τy extends to a continuous linear mapping from C∞ℝd into itself(3)
Iff,g∈Sℝd and y∈ℝd, then(20)∫ℝdτyfxgxϑkxdx=∫ℝdfxτ−ygxϑkxdx.
The generalized translation operator is used to define a convolution structure: Forf,g∈Sℝd,(21)f∗kgx≔ck−1∫ℝdfyτxg˘yϑkydy,where g˘x≔g−x. We can also write the convolution ∗k as
(22)f∗kgx=ck−1∫ℝdFkfξFkgξEkix,ξϑkξdξ.
We refer the reader to [15] for more details on the convolution product ∗k. It is worth mentioning, for the distributional version of the Dunkl transform, the translation operator and the Dunkl convolution of distributions and properties, we refer the reader to [30].
Forn∈ℕ, let Hkn be the space of k-harmonic polynomials of degree n on ℝd,
(23)Hkn=KerΔk∩Pnℝd,where Δk is the Dunkl Laplacian and Pnℝd denotes the space of homogeneous polynomials of degree n on ℝd. The restriction of elements in Hkn on the unit sphere Sd−1 in ℝd are the so-called k-spherical harmonics. We shall not distinguish between Ykn∈Hkn and its restriction to Sd−1. The space Hkn has a reproducing kernel Qkn·,· in the sense that
(24)fx=dk−1∫Sd−1fyQknx,yϑkydσy,∀f∈Hkn,∥x∥≤1.
Here,dk is the constant
(25)dk≔∫Sd−1ϑkxdσx=ck/2λkΓλk+1,where ck and λk are as defined in (13) and (16), respectively. According to [31], for x,y≠0, the kernel Qkn can be written as
(26)Qknx,y=∥x∥∥y∥nn+λk/λkVkCnλk·,y/∥y∥x/∥x∥,where Vk is the Dunkl intertwining operator (8), and Cnα is the Gegenbauer polynomial of degree n,(27)Cnαx=2αn/n!2F1−n,n+2α,α+12;1−x/2,for α>0, with 2F1 is the Gauss hypergeometric function.
The following analogue of the Funk-Hecke formula fork-spherical harmonics will be used later on; for the proof, the reader is referred to [32]. Let h be a continuous function on −1,1. Then, for any Ykn∈Hkn,(28)1dk∫Sd−1Vkhx,·yYknyϑkydσy=ΛnhYknx,x∈Sd,where Λnh is a constant defined by
(29)Λnh≔Γλk+1/πΓλk+12n!/2λkn∫−11htCnλkt1−t2λk−1/2dt.
We summarize some basic properties of Gegenbauer polynomials in a way that we shall use later.Fact 3 (see [[6] (1.2.10)], [[33], (3.32.3)]).
Forλ∈ℂ such that Reλ>0, the following two integral formulas hold:
(30)∫−111−t2λ−1/2CmλtCnλtdt=λ/n+λπΓλ+1/2Γ2λ+n/n!Γλ+1Γ2λδm,n.∫−11eizt1−t2λ−1/2Cnλtdt=πinΓλ+1/2Γ2λ+n/n!Γ2λΓλ+n+1z/2njλ+nz.
LetDRℝe denote the space of even compactly supported smooth functions with support in −R,R, where R>0. The Paley-Wiener theorem for the Hankel transform Hα (see (17)) states that Hα maps DRℝe bijectively onto the space HRℂe of even entire functions g satisfying, for all N∈ℕ,(31)∣gz∣≤CN1+∣z∣−NeR∣Imz∣,∀z∈ℂ,for some positive constant CN; see for instance [34].
This result has been generalized by de Jeu [23] to the Dunkl transform. To state the (complex) Paley-Wiener theorem for Fk, we introduce the following notation. For R>0, let HRℂd be the space of entire functions F on ℂd with the property that for all N∈ℕ there exists a constant CN>0 such that
(32)∣Fz∣≤CN1+∥z∥−NeR∥Imz∥,∀z∈ℂd.
We letDRℝd denote the space of smooth compactly supported functions with support contained in the closed ball BO,R¯⊂ℝd with radius R>0 and the origin as center.Fact 4 (see [23]).
The Dunkl transformFk is a linear isomorphism between DRℝd and HRℂd, for all R>0.An immediate consequence of the above Paley-Wiener theorems can be stated as follows:Lemma 5.
LetF0∈C∞ℝe.Then, F0∥ξ∥=Fkfξ for some radial function finDRℝd if and only if F0 extends to an entire function on ℂ satisfying the estimate
(33)∣F0z∣≤CN1+∣z∣−NeRImz,∀z∈ℂ,for all N∈ℕ.Proof.
The statement follows from the fact thatFkfξ=Hλkf0∥ξ∥ whenever f is a radial function with fx=f0∥x∥ (see (15)), together with the Paley-Wiener theorems stated above for the Hankel and the Dunkl transforms.


## 3. The Range of the Spectral Projection Associated withΔk

Recall from (12) that the Dunkl transform of f∈Dℝd is defined by
(34)Fkfξ=ck−1∫ℝdfxEkx,−iξϑkxdx.Using polar coordinates, the Dunkl inversion formula (14) becomes
(35)fx=ck−1∫0∞s2λk+1∫Sd−1FkfsηEkix,sηϑkηdσηds=∫0∞s2λk+1Pksfxds,where
(36)Pksfx≔ck−1∫Sd−1FkfsηEkix,sηϑkηdση.Notice thatΔkxPksfx=−s2Pksfx.From (36), we may derive a second formula for Pksf. Indeed, substituting (34) into (36), we obtain
(37)Pksfx=ck−2∫Sd−1∫ℝdfyEk−isη,yϑkydyEkix,sηϑkηdση=ck−2∫ℝdfy∫Sd−1Ekη,−isyEkisx,ηϑkηdση⏟≔Ikx,y;sϑkydy.According to [[35], page 2424], the inner integral is equal to
(38)Ikx,y;s=dk∑n=0∞Γλk+1/2nΓλk+n+12jn+λks∥x∥jn+λks∥y∥Qknisx,−isy,where dk is the constant (25), Qkn·,· is the reproducing kernel (26), and jα is the normalized Bessel function (18). We now use the well-known addition formula for Bessel functions (see [[36] , p. 215]): for a,b,θ∈ℝ,(39)jαa2+b2−2abcosθ=∑n=0∞n+α/αΓα+1/2nΓα+n+12abnjn+αajn+αbCnαcosθ,which converges uniformly with respect to θ∈ℝ. Using (38) and (26) together with the Laplace representation (9) for Vk, we deduce that
(40)Ikx,y;s=dk∫ℝdjλks∥x∥2+∥y∥2−2∥y∥x,ηdμy/∥y∥η=dk∫ℝdjλks∥x∥2+∥y∥2−2x,ηdμyη=dkVkjλks∥x∥2+∥y∥2−2x,·y.Definejs,λky≔jλks∥y∥, then, by [[35] , p. 2429], we have
(41)Vkjλks∥x∥2+∥y∥2−2x,·y=τyjs,λk−x=τ−xjs,λky.Consequently, the eigenfunctionPksf can be rewritten as
(42)Pksfx=dkck−2∫ℝdτxfyjλks∥y∥ϑkydy(43)Pksfx=2−λkΓλk+1−1f∗kjs,λky.Above, we have used some of the properties of the generalized translation operator listed in Fact2. The following statement lists the necessary conditions for Theorem 9.Proposition 6.
Assume thatf∈DRℝd and let Pksfx defined either by (36) or (42), with s∈ℝandx∈ℝd. Then the following hold:
(1)
Pksfx is a smooth function on ℝ×ℝd(2)
ΔkxPksfx=−s2Pksfx, where the upper index x indicates the relevant variable(3)
Forx∈ℝd be given, Pksfx extends to an even entire function of s∈ℂ(4)
For everyN∈ℕ and for every multi-index m∈ℕd, there exists a constant Ck,m,N>0 such that(44)∣∂xmPksfx∣≤Ck,m,N1+∣s∣−NeR+∥x∥∣Ims∣,∀s∈ℂ(5)
For anyk-spherical harmonic Ykℓ of degree ℓ and for every r>0, the map(45)s↦s−2ℓ∫Sd−1PksfrωYkℓωϑkωdσωis entire on ℂ.Proof.
(1)
In view of properties of the translation operatorτx and the normalized Bessel function jα, the first statement follows from the representation (42) of Pksfx.(2)
The second property is immediate from (36), since ΔkxEkx,isη=−s2Ekx,isη.(3)
Now, by (42), the map s↦Pksf is certainly extends to an even entire function on ℂ(4)
Sincef∈DRℝd, it follows from the Paley-Wiener theorem for the Dunkl transform that Fkf extends to an entire function on ℂd satisfying the estimate(46)∣Fkfξ∣≤Ck,M1+∥ξ∥−MeR∥Imξ∥for all M∈ℕ; see Fact 4. Consequently, by (36), we obtain
(47)∣∂xmPksfx∣≤Ck,M′1+∣s∣−Ms∣m∣eR+∥x∥∣Ims∣for all M∈ℕ, where we used the estimate ∣∂xmEkx,z∣≤∥z∥∣m∣e∥x∥∥Rez∥ for all x∈ℝd and z∈ℂd. This finishes the proof of the estimate (44).
(5)
LetYkℓ be a k-spherical harmonic of degree ℓ. By the Fubini theorem and (37), we have(48)∫Sd−1PksfrωYkℓωϑkωdσω=ck−2∫ℝdfy∫Sd−1Ikrω,y;sYkℓωϑkωdσωϑkydy,where Ikrω,y;s is as in (38),
(49)Ikrω,y;s=dk∑n=0∞n+λk/λkΓλk+1/2nΓλk+n+12rs2∥y∥njn+λkrsjn+λks∥y∥VkCnλk·,y/∥y∥ω.
We now apply the Funk-Hecke formula (28) to deduce that
(50)∫Sd−1VkCnλk·,y/∥y∥ωYkℓωϑkωdσω=dkΛℓCnλkYkℓy/∥y∥,where, by Fact 3, we have
(51)ΛℓCnλk=Γλk+1/πΓλk+12ℓ!/2λkℓ∫−11CnλktCℓλkt1−t2λk−1/2dt=λk/λk+ℓδℓ,n.
Using the above identities, it follows that(52)∫Sd−1PksfrωYkℓωϑkωdσω=122λk+2ℓΓλk+ℓ+12∫ℝdfyrs2∥y∥ℓjℓ+λkrsjℓ+λks∥y∥Ykℓy/∥y∥ϑkydy=φℓ+λkrs∫0∞φℓ+λkstfℓtt2λk+1dt,where φℓ+λkz≔1/2λk+ℓΓλk+ℓ+1zℓjℓ+λkz and
(53)fℓt≔∫Sd−1ftηYkℓηϑkηdση.
Above, we have used the fact thatdk=ck/2λkΓλk+1; see (25). In conclusion,
(54)s−2ℓ∫Sd−1PksfrωYkℓωϑkωdσω=s−ℓφℓ+λkrs∫0∞s−ℓφℓ+λkstfℓtt2λk+1dt.
The desired result now follows from the fact thats−ℓφℓ+λksz is an entire function of s∈ℂ.The following lemma is needed for later use.Lemma 7.
(1)
For any radial functionf on ℝd with fx=f0∥x∥ and for any k-spherical harmonic Ykℓ of degree ℓ, we have(55)ΔkfYkℓ=f0″r+2λk+1/rf0′r−ℓℓ+2λk/r2f0rYkℓω.(2)
Under the change of variablefr=rc−αur/c, the differential equation(56)f′′r+A/rf′r+μ2−B/r2fr=0transforms to the Bessel equation u′′r+1/ru′r+1−ν2/r2ur=0 with
(57)A=2α+1,μ=c−1,B=ν2−α2.Proof.
(1)
In the polar coordinatesx=rω, the Dunkl Laplacian operator is expressed as(58)Δk=d2dr2+2λk+1rddr+1r2Δk,Sd−1,where Δk,Sd−1 is the analogue of the Laplace-Beltrami operator on the sphere Sd−1, which, in particular, has k-spherical harmonics as eigenfunctions,
(59)Δk,Sd−1Ykℓω=−ℓℓ+2λkYkℓω.
We refer the reader to [37] for more details on Δk,Sd−1.(2)
Obvious
Next we will list the sufficient conditions for Theorem9.Proposition 8.
Fors∈ℝ and x∈ℝd,letfsx be a function satisfying the following conditions:
(1)
fsx is smooth on ℝ×ℝd(2)
fsx is an eigenfunction of the Dunkl Laplacian with eigenvalue −s2(3)
The mappings↦fsx extends to an even entire function on ℂ(4)
For everyN∈ℕ and for every multi-index m∈ℕd, there exists a positive continuous increasing function Ck,m,N on ℝ+ such that(60)∣∂xmfsx∣≤Ck,m,N∥x∥1+∣s∣−NeR+∥x∥∣Ims∣,∀s∈ℂ(5)
For everyr>0 and every k-spherical harmonic Ykℓ of degree ℓ, the mapping(61)s↦s−2ℓ∫Sd−1fsrωYkℓωϑkωdσωis entire on ℂ.
Then, there existsf∈DRℝd such that fsx=Pksfx.Proof.
Define the following function(62)fx≔∫0∞s2λk+1fsxds.
The estimate (60) (with m=0) shows that the integral converge absolutely, and therefore, by assumption (1) and the estimate (60) again, f is smooth on ℝd.
Below, we will prove thatfx=0 for all ∥x∥>R. Let Yℓ,j:1≤j≤dimHkℓ be an orthonormal basis of Hkℓ. By (62), we have
(63)fℓ,jr=∫0∞s2λk+1fs,ℓ,jrds,r>0,where fℓ,jr≔fr·,Yℓ,jk and fs,ℓ,jr≔fsr·,Yℓ,jk are the k-spherical harmonic coefficients of f and fs, respectively. Here, ·,·k stands for the inner product in L2Sd−1,ϑkηdση. To show that fx=0 for all ∥x∥>R, it is enough to prove that fℓ,jr=0 for all r>R. By Lemma 7 (1), the fact that fs,ℓ,jrYℓ,jη are eigenfunctions of Δk with eigenvalue −s2 implies
(64)d2dr2+2λk+1rddr+s2−ℓℓ+2λkr2fs,ℓ,jr=0.
Further, by Lemma7 (2), the solution fs,ℓ,jr of equation (64) is in the following form:
(65)fs,ℓ,jr=cℓ,jsrℓjℓ+λkrs=c~ℓ,jsrℓs2ℓjℓ+λkrs,where cℓ,j is a function which depends only on s, and c~ℓ,js≔s−2ℓcℓ,js. Because of the condition (5) on fsx, the mapping s↦c~ℓ,js extends to an entire function of s∈ℂ.
On the other hand, it is known that the normalized Bessel functionjαz has infinitely many positive zeros 0<ρα1<ρα2<⋯. Let 0<r0<ρℓ+λk1 and define
(66)ir0≔infz∈ℂ,∣z∣=r0jℓ+λkz.
Fors∈ℂ such that ∣s∣>0, the identity (65) and the assumption (4) on fsx (with m=0) imply, for r=r0/∣s∣,(67)∣c~ℓ,js∣≤Ck,0,Nr0sir0−1r0−ℓs−ℓ1+s−NeR+r0sIms,for all N∈ℕ. In particular, if ∣s∣≥1, we have
(68)∣c~ℓ,js∣≤Ck,M,r01+s−MeR+r0Ims,for all M∈ℕ (recall that Ck,0,N is an increasing function). For the compact domain 0≤∣s∣≤1, the estimate (68) holds true with a different constant. Moreover, by (65) and the evenness of fsx, the map s↦c~ℓ,js is even. Applying Lemma 5, there exists a radial function ψ∈DR+r0ℝd+2ℓ such that c~ℓ,js=Hk+d+2ℓ−2/2ψ0∥ξ∥ with ∥ξ∥=s, where ψx=ψ0x and Hα is the Hankel transform. Here, we have used the fact that the Dunkl transform of radial functions at ξ is a Hankel transform at ∥ξ∥ (see (15)). Now, letting r0⟶0 show suppψ⊂BO,R¯ in ℝd+2ℓ.
Using again (65) and the fact that c~ℓ,js=Hk+d+2ℓ−2/2ψ0s, the integral (63) becomes
(69)fℓ,jr=rℓ∫0∞c~ℓ,jsjℓ+λkrss2ℓ+2λk+1ds=rℓ∫0∞Hk+d+2ℓ−2/2ψ0sjℓ+λkrss2ℓ+2λk+1ds=rℓ∫0∞Hλk+ℓψ0sjℓ+λkrss2ℓ+2λk+1ds=2λk+ℓΓλk+ℓrℓψ0r.
That is,(70)ψx=2−λk+ℓΓλk+ℓ−1∥x∥−ℓfℓ,j∥x∥,x∈ℝd+2ℓ,which implies that fℓ,j∥x∥=0 for ∥x∥>R as desired.
It will follow thatfsx=Pksfx with fx=∫0∞s2λk+1fsxds provided we prove that if a function hsx satisfies assumptions (1)–(5) of Proposition 8 with ∫0∞s2λk+1hsxds=0, then hsx=0 for all x∈ℝd and s∈ℝ. To prove so, it suffices to show that ∫0∞s2λk+1hs,ℓ,jxds=0 implies hs,ℓ,jx=0 for all x∈ℝd and s∈ℝ. However, this follows by mimicking the proof given above together with the injectivity of the Hankel transform on ℝ+.We can now state the main result of this paper by putting the above propositions together.Theorem 9.
Letx∈ℝd and s∈ℝ. There exists a smooth compactly supported function f with support contained in BO,R¯ such that Pksfx=fsx if and only if fsx satisfies the conditions listed in Proposition 8.The following statement illustrates an interesting application of Theorem9. We may think of it as a spectral-reformulation of the Paley-Wiener theorem for the Dunkl transform (see Fact 4).Theorem 10.
Letφs,η be a smooth function on ℝ×Sd−1.Then, φs,η=Fkfsηfor somef∈DRℝd if and only if the following two conditions hold:
(1)
For eachη∈Sd−1, the map s↦φs,η has an entire extension with the property that for all N∈ℕ, there exists a constant Ck,N>0 such that(71)∣φs,η∣≤Ck,N1+∣s∣−NeR∣Ims∣(2)
For an arbitraryk-spherical harmonic Ykℓ of degree ℓ, the map(72)s↦s−ℓ∫Sd−1φs,ηYkℓηϑkηdσηhas an even entire extension.Proof.
Recall from (10) the Dunkl kernel Ekx,y=Vke·,yx. Following [38], let tVk be the transpose of the Dunkl intertwining operator Vk which satisfies on Dℝd(73)∫ℝdfxVkhxϑkxdx=∫ℝdtVkfxhxdx.
By [38], suppf⊂BO,R¯ if and only if supptVkf⊂BO,R¯.
We now proceed towards the proof of the direct part. Assume thatφs,η=Fkfsη for some f∈DRℝd. Then,
(74)φs,η=∫ℝdfxVke−isη,·xϑkxdx=∫ℝgηpe−ispdp,where gηp≔∫η,y=ptVkfydy. Since suppf⊂BO,R¯, it follows that gη∈DRℝ. This together with the Paley-Wiener theorem for the Euclidean Fourier transform on ℝ completes the proof of the property (1).
Next, we turn our attention to the property (2). Forr>0 and s∈ℂ, we have
(75)∫Sd−1PksfrηYkℓηϑkηdση=ck−1∫Sd−1Ykℓη∫Sd−1FkfsωEkω,irsηϑkωdσωϑkηdση=ck−1∫Sd−1Fkfsω∫Sd−1Ekirsω,ηYkℓηϑkηdσηϑkωdσω.
Using the Funk-Hecke formula (28) to deduce that
(76)∫Sd−1Ekirsω,ηYkℓηϑkηdση=dkΛℓeirs·Ykℓω,where, by Fact 3,
(77)Λneirs·=Γλk+1/πΓλk+12ℓ!/2λkℓ∫−11eirstCℓλkt1−t2λk−1/2dt=Γλk+1/2ℓΓλk+ℓ+1irsℓjℓ+λkrs.
This shows that(78)∫Sd−1PksfrηYkℓηϑkηdση=1/2λk+ℓΓλk+ℓ+1irsℓjℓ+λkrs∫Sd−1FkfsωYkℓωϑkωdσω.
That is,(79)s−2ℓ∫Sd−1PksfrηYkℓηϑkηdση=1/2λk+ℓΓλk+ℓ+1irℓjℓ+λkrss−ℓ∫Sd−1φs,ωYkℓωϑkωdσω.
Sinces↦Pksf and s↦jℓ+λkrs are even, it follows that the map s↦s−ℓ∫Sd−1φs,ωYkℓωϑkωdσω is even. The analycity of the latter map is immediate from Preposition 6.
For the converse part, define(80)fsx≔ck−1∫Sd−1φs,ηEkix,sηϑkηdση,s∈ℝ,x∈ℝd.
By the assumptions (1) and (2) onφs,η and (11), the function fs satisfies the hypothesis of Theorem 9. Thus, there exists f∈DRℝd such that fsx=Pksfx. That is,
(81)ck−1∫Sd−1φs,η−FkfsηEkix,sηϑkηdση=0,which implies
(82)ck−1∫0∞∫Sd−1φs,η−FkfsηEkix,sηϑkηdσηs2λk+1ds=0.
This finishes the proof of the converse part.



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*Source: 7803719-2020-07-24.xml*