Dual of 2D Fractional Fourier Transform Associated to It\^ O--Hermite Polynomials

$Dual of 2D Fractional Fourier Transform Associated to It\^ O--Hermite Polynomials$

DUAL OF 2D FRACTIONAL FOURIER TRANSFORM ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS

ABDELHADI BENAHMADI AND ALLAL GHANMI

Dedicated to the memory of Professor Elhachmia Ait Benhaddou

Abstract. A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô– Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô–Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established

1 Introduction The role played by the classical Mehler formula [14], +∞ tnH (x)H (y) 1 −t2(x2 + y2) + 2txy ! X n n = √ exp , (1.1) n 2 2 n=0 2 n! 1 − t 1 − t n x2 dn −x2 for the real Hermite polynomials Hn(x) := (−1) e dxn (e ), is well known in the literature [14, 18, 3, 12, 15, 17, 13, 8, 16]. Its complex analogues for Itô–Hermite ν polynomials Hm,n have been obtained in [19, 4, 10, 6] and have been employed in [6] ν to establish integral reproducing property for Hm,n by a like Fourier transform, and to provide a closed expression of the heat kernel for the Cauchy initial value problem for 2 2,ν −ν|z|2 a special magnetic Laplacian acting on the Hilbert space Lν(C) := L (C; e dλ). In [20], Zayed has used the one in (2.1) below to construct a non trivial 2d fractional Fourier transform Z ν ν −ν|ζ|2 Fu,vψ(ξ) = ψ(ζ)Ku,v(ζ; ξ)e dxdy; ζ = x + iy, (1.2) C ν whose eigenfunctions are the Itô–Hermite polynomials. Here Ku,v(ζ; ξ) stands for the kernel function ν ν n o arXiv:2005.08490v1 [math.CV] 18 May 2020 Kν (ζ; ξ) = exp −uv(|ζ|2 + |ξ|2) + uζξ + vζξ . (1.3) u,v π(1 − uv) 1 − uv In the present paper, we explore further applications in the context of integral 2 2 2 transforms and weighted Bergman spaces Bα,β(D ) on the bi-disk D = D × D; D = {z ∈ C, zz < 1}, deﬁned as the Hilbert space of all analytic functions on D2 that are square integrable with respect to the measure 2 2 dµα,β(z, w) = ωα,β(|z| , |w| )dλ(z, w), (1.4)

2010 Mathematics Subject Classiﬁcation. Primary 44A20; 30G35; 30H20 Secondary 47B38; 30D55. Key words and phrases. Weighted Bergman space on bi-disk; Itô–Hermite polynomials; Mehler formula; Singular values; p-Schatten class; Segal–Bargmann transform. 1 2 ABDELHADI BENAHMADI AND ALLAL GHANMI where the weight function is given by α β ωα,β(s, t) := (1 − s) (1 − t) ; α, β > −1, (1.5) and dλ denotes the standard Lebesgue measure. To this end we follow the scheme already applied in [7] to introduce and study the dual transforms of fractional Hankel transforms with ranges in weighted Bergman space on the disk. Mainly, we consider the family of integral transforms Z ν ν −ν|z|2 Rwf(u, v) = f(z)Ku,v(z; w)e dλ(z) (1.6) C on D2, labeled by ν > 0 and w ∈ C and seen as the dual transform of the 2d fractional ν ν Fourier transform in (1.2), Rwf(u, v) = Fu,vf(w). The aim in this paper concern identiﬁcation of the null space and the range of the ν transforms Rw. We also study their boundedness and provide complete description of their compactness and membership in p-Schatten class. Our main results can be stated as follows

ν 2 Theorem 1.1. The integral transform Rw is well defined and bounded from Lν(C) 2 2 into the weighted Bergman space Bα,β(D ) if and only if α > 0 and β > 0. The ν characterization of its null space Ker(Rw) depends on the zeros set of Itô–Hermite ν ν polynomials Hm,n. Namely, if Nw(H) = {(m, n); m, n = 0, 1, 2, ··· ; Hm,n(w, w) = 0}, ν then Ker(Rw) is a vector space spanned as ν ν Ker(Rw) = Span{Hm,n;(m, n) ∈ Nw(H)}. ν 2 2 2 Theorem 1.2. For α, β > 0, the operator Rw : Lν(C) −→ Bα,β(D ) is compact and its singular values are given by νπΓ(α + 1)Γ(β + 1) !1/2 sν,α,β(w) = |Hν (w, w)|. m,n νm+nΓ(α + m + 2)Γ(β + n + 2) m,n Moreover, it belongs to the p-Schatten class for every p > max(2/(α + 1); 2/(β + 1)). The proof of Theorem 1.1 is contained in Propositions 3.1, 3.2, 3.3 and 3.5 presented in Section 3, while the one of Theorem 1.2 is given in Section 4. The next section is devoted to some preliminaries concerning weighted Bergman space on the bi-disk and Itô–Hermite polynomials. We conclude the paper by discussing the close connection of ν Rw to the fractional Hankel transforms √ α/2 ! 2 2 2ν u Z ∞ 2ν uv −ν(x +uvy ) ν,α 1−uv Hu,v (f)(y) = xf(x)Iα xy e dr, (1.7) 1 − uv v 0 1 − uv where Iα denotes the modified Bessel function [1, p.222]. 2 Theorem 1.3. Let f ∈ Lν(C) and gk the associated Fourier coefficients. Then, ν,α ν Hu,v (gk), for varying integer k, are the fractional Fourier coefficients of Fu,vf, the fractional Fourier transform of f.

2 Preliminaries 2 2 For ﬁxed reals α, β > −1, the weighted Bergman space Bα,β(D ) is a closed subspace 2 2 2 of the Hilbert space Lα,β(D ) := L (D × D; dµα,β) endowed with the scaler product Z hf, giα,β = f(z, w)g(z, w)dµα,β(z, w). D2 DUAL OF 2D FRFT ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS 3

2 2 We denote by k·kα,β the associated norm. An orthonormal basis of Bα,β(D ) is given by α,β α,β −1/2 ϕm,n := γm,n em,n m n α,β where em,n(z, w) := z w and γm,n is its square norm given by π2Γ(α + 1)Γ(β + 1)m!n! γα,β := = ke k2 . m,n Γ(α + m + 2)Γ(β + n + 2) m,n α,β 2 2 Subsequently, the sequential characterization of Bα,β(D ) is given by  ∞ ∞  2 2  X X α,β 2  Bα,β(D ) = am;nem,n; γm,n|am;n| < ∞ . m,n=0 m,n=0  2 2 2 2 Accordingly, B0,0(D ) is identified to be the Hardy space on the bi-disk, while B−1,−1(D ) 2 2 2 is the classical Bergman space on D . The reproducing kernel of Bα,β(D ) is given by (α + 1)(β + 1) K ((u, v); (z, w)) = . α,β π2(1 − uz)α+2(1 − vw)β+2 2 2 Thus Bα,β(D ) is obtained as the Bergman projection (α + 1)(β + 1) Z ϕ(z, w) P (ϕ)(u, v) = dµα,β(z, w) π2 D2 (1 − uz)α+2(1 − vw)β+2 2 2 of Lα,β(D ). Another realization is by means of the unitary two-dimensional second Bargmann transform Z ! 1 α β sw + tz − (s + t) Bα,βϕ(z, w) = α+1 β+1 s t exp ϕ(s, t)dsdt (1 − z) (1 − w) R+2 (1 − z)(1 − w) 2 +2 α β −x−y acting on the Hilbert space L (R ; x y e dxdy). The kernel function of Bα,β ap- pears as the tensor product of two copies of the kernel function of the standard one- dimensional second Bargmann transform [2, p. 203]. However, it can be seen as the bilinear generating function involving the product of generalized Laguerre polynomials (α) (β) Lm (s)Ln (t). 2 2 In the sequel, we will provide interesting realization of specific subspaces of Bα,β(D ) by invoking the complex Mehler function 2 0 2 0 ! 0 1 −uv(ν|z| + ν |w| ) + ν uzw + νvzw Kν;ν (z, w) = exp , (2.1) u,v 1 − uv 1 − uv 0 ν with ν, ν > 0 and u, v ∈ D, associated to Itô–Hermite polynomials Hm,n defined on the complex plane C by [11, 9, 5] ∂m+n Hν (z, z) = (−1)m+neνzz e−νzz . (2.2) m,n ∂zm∂zn ν 2 To this end, let recall that the Hm,n form an orthogonal basis of Lν(C) and that the Mehler function in (2.1) can be expanded in terms of normalized Itô–Hermite polynomials 1/2 ν 0 ψν := Hν (2.3) m,n πνm+nm!n! m,n as [6] ∞ ν;ν0 X m n ν ν Ku,v (z, w) := u v ψm,n(z)ψm,n(w). (2.4) m,n=0 4 ABDELHADI BENAHMADI AND ALLAL GHANMI

For ν = ν0 = 1, this is exactly the one announced by Wünsche [19] and proved later by Ismail in [10, Theorem 3.3] as a speciﬁc case of his Kibble–Slepian formula [10, Theorem 1.1].

ν 3 Basic properties of Rw We begin by observing that the kernel function in (1.3) reads in terms of the one in (2.1) as ν Kν (z, w) = Kν;ν(z, w). (3.1) u,v π u,v and therefore satisfies Z ν ν −ν|z|2 ν Ku,v(z; w)Ku,v(z; w)e dλ(z) = K|u|2,|v|2 (w; w) > 0. (3.2) C ν 2 Proposition 3.1. The integral transform Rw is well defined on Lν(C). Proof. Using (3.2) and the Cauchy-Schwarz inequality, we obtain Z 1/2 ν ν 2 |R f(u, v)| ≤ |K (w; w)| kfk 2 w u,v Lν (C) C ν 1/2 ≤ K 2 2 (w; w) kfk 2 (3.3) |u| ,|v| Lν (C) 2 for every f ∈ Lν(C). ν ν The action of Rw on ψm,n in (2.3) is given by ν ν ν Rwψm,n = ψm,n(w)em,n. ν ν This follows by means of (3.1) and (2.4). Therefore, the family Rwψm,n, for varying 2 2 m, n, form an orthogonal system in Lα,β(D ) since the monomials em,n are. The next ν 2 result discusses the boundedness of Rw from Lν(C) into the weighted Hilbert space 2 2 Lα,β(D ). ν 2 Proposition 3.2. For α > 0 and β > 0, the operator Rw is bounded from Lν(C) into 2 2 Lα,β(D ). Proof. Set Z ν,α,β ν kw := K|u|2,|v|2 (w; w)dµα,β(u, v). (3.4) D2 Then, from (3.3), we have ν 2 ν,α,β 2 kR fk ≤ k kfk 2 . (3.5) w α,β w Lν (C) ν ν,α,β Subsequently, the boundedness of the operator Rw requires that kw be finite. But, ν using the closed expression of K|u|2,|v|2 (w; w), we get Z 1 Z 1 ! ν,α,β ν(s + t − 2st) 2 ωα,β(s, t) kw = νπ exp |w| dsdt. (3.6) 0 0 1 − st 1 − st Hence since 0 ≤ (s + t − 2st)/(1 − st) ≤ 1 and 1/(1 − st) ≤ 1/(1 − s)(1 − t), it follows Z 1 Z 1 Z 1 Z 1 ν,α,β ν|w|2 νπ ωα,β(s, t)dsdt ≤ kw ≤ νπe ωα−1,β−1(s, t)dsdt. (3.7) 0 0 0 0 ν ν 2 2 Thus, Rw is bounded for α, β > 0. In this case Rwf belongs to Lα,β(D ) for every 2 f ∈ Lν(C). DUAL OF 2D FRFT ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS 5

ν 2 Now, appealing to the fact that ψm,n constitutes an orthonormal basis of Lν(C), we ∞ 2 P ν can expand any f ∈ Lν(C) as f = αm,nψm,n, so that one gets m,n=0 ∞ ν X ν α,β 1/2 α,β Rwf = αm,nψm,n(w) γm,n ϕm,n. (3.8) m,n=0 The series in (3.8) converges uniformly on compact sets of the complex plane. Direct computation shows that we have ∞ ν 2 X 2 ν 2 α,β kRwfkα,β = |αm,n| |ψm,n(w)| γm,n. (3.9) m,n=0 ν Accordingly, the description of the range and the null space of the Rw are closely ν connected to zeros of Itô–Hermite polynomials. Thus, we let Z(Hm,n) denotes the zeros ν ν set of Hm,n for fixed m, n, while Z(H) := ∪m,nZ(Hm,n). We also set ν Nw(H) = {(m, n); m, n = 0, 1, 2, ··· ; Hm,n(w, w) = 0}. ν Proposition 3.3. Let α > 0 and β > 0. If w∈ / Z(H), then the range of Rw acting on 2 2 2 Lν(C) is a specific subspace of the weighted Bergman space Bα,β(D ). ν Proof. This is immediate by means of (3.8) and Proposition 3.2. Indeed, Rwf belongs 2 2 2 to Lα,β(D ) and is clearly holomorphic on D by Stone-Weierstrass theorem. Hence, ν 2 2 2 Rw(Lν(C)) ⊂ Bα,β(D ). Remark 3.4. Concerning the converse inclusion, we can provide an explicit example 2 ν 2 2 showing that the range of Lν(C) by Rw is strictly contained in Bα,β(D ). However, ν this can be reproved using compactness (discussed below) of the transform Rw, since the 2 2 range of compact operator is not closed in Bα,β(D ) ay least for in α, β > 0. ν 2 Proposition 3.5. The null space of Rw acting on Lν(C) is a vector space characterized explicitly as ν Ker(Rw) = Span{Hm,n;(m, n) ∈ Nw(H)} ν with dimension equals to the cardinal of Nw(H). Thus, the integral transform Rw is one-to-one if and only if w∈ / Z(H). ∞ P ν 2 ν Proof. According to (3.9), if f = αm,nψm,n ∈ Lν(C) is in the null space of Rw, m,n=0 ν ν then kRwfk = 0 and hence αm,nψm,n(w) = 0 for every m, n. Therefore, the null space ν of Rw reduces to f = 0 when w∈ / Z(H). Now, for w ∈ Z(H), we conclude that P ν αm,n = 0 for all (m, n) ∈/ Nw(H). Therefore, f = αm,nψm,n which proves (m,n)∈Nw(H) ν ν Ker(Rw) ⊂ Span{ψm,n;(m, n) ∈ Nw(H)}. ν The converse inclusion is trivial and hence dim(Ker(Rw)) = Cardinal(N(H)). 4 Proof of Theorem 1.2: Compactness and membership in p-Schatten class Set ν,α,β ν α,β 1/2 cm,n (w) := ψm,n(w) γm,n and consider the finite rank operators p q X X ν,α,β D ν E α,β Rp,qf = c (w) f, ψ ϕ m,n m,n L2 ( ) m,n m=0 n=0 ν C 6 ABDELHADI BENAHMADI AND ALLAL GHANMI

ν,α,β 2 ν|w|2 α,β which are bounded and compact. Then, using the fact |cm,n (w)| ≤ e γm,n as well ∞ 2 P D ν E 2 as kfk 2 = | f, ψ | , we obtain Lν (C) m,n 2 m,n=0 Lν (C) ∞ ∞ 2 ν 2 X X ν,α,β 2 D ν E k(R − Rp,q)fk = c (w) f, ψ w m,n m,n L2 ( ) m=p+1 n=q+1 ν C  ∞ ∞  X X ν,α,β 2 2 ≤  |c (w)|  kfk 2 m,n Lν (C) m=p+1 n=q+1  ∞ ∞  ν|w|2 X X α,β 2 ≤ e  γ  kfk 2 , m,n Lν (C) m=p+1 n=q+1 so that the following estimate for the operator norm ∞ ∞ ν 2 ν|w|2 X X α,β kRw − Rp,qk ≤ e γm,n. m=p+1 n=q+1 follows. The series in the right-hand side is convergent for α > 0 and β > 0, and hence ν its rest goes to zero, so that lim kRp,q − R k = 0. p,q−→∞ w The above discussion can be reformulated as follows.

ν Proposition 4.1. Let α > 0 and β > 0, then Rw(f) is compact. Remark 4.2. The expansion in (3.8), ∞ D E Rν (f) = X cν,α,β(w) f, ψν ϕα,β , w m,n m,n L2 ( ) m,n m,n=0 ν C ν looks like the spectral decomposition of the operator Rw. ν From general context, we know that the adjoint of the integral transform Rw is given through Z ν ∗ D ν E ν (Rw) g(z) = g, Ku,v(·; w) = g(u, v)Ku,v(z; w)dµα,β α,β D2 2 2 for every function g ∈ Lα,β(D ). This can easily be handled by direct computation. ν ∗ ν 2 Subsequently, (Rw) Rw is an integral transform on Lν(C) Z ν ∗ ν ν −ν|ζ|2 (Rw) Rwf = Sw(ζ, z)f(ζ)e dλ(ζ) C with kernel function given by ∞ ν X ν,α,β 2 ν ν Sw(ζ, z) := |cm,n (w)| ψm,n(z)ψm,n(ζ). m,n=0 Therefore, we have ∞ ν ∗ ν X ν,α,β 2 D ν E ν (Rw) Rw(f) = |cm,n (w)| f, ψm,n ψm,n, m,n=0 and in particular ν ∗ ν ν ν,α,β 2 ν (Rw) Rw(ψm,n) = |cm,n (w)| ψm,n. ν 2 ν Accordingly, since ψm,n form an orthonormal basis of Lν(C), the singular values of Rw ν ν ∗ ν 1/2 which are the eigenvalue of |Rw| := ((Rw) Rw) are given by

ν,α,β ν,α,β ν α,β 1/2 sm,n (w) = |cm,n (w)| := |ψm,n(w)| γm,n . DUAL OF 2D FRFT ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS 7

More explicitly, νπΓ(α + 1)Γ(β + 1) !1/2 sν,α,β(w) = |Hν (w, w)|. (4.1) m,n νm+nΓ(α + m + 2)Γ(β + n + 2) m,n Subsequently,

1 2 ! 2 ν,α,β ν|w| m!n!Γ(α + 1)Γ(β + 1) s (w) ≤ πe 2 . m,n Γ(m + α + 2)Γ(n + β + 2) It follows that lim sν,α,β(w) = 0 since α, β > 0 and the right hand-side behaves m,n−→∞ m,n −α−1 −β−1 ν as m n for m, n large enough. Moreover, Rw is in the p-Schatten class if p(α + 1)/2 > 1 and p(β + 1)/2 > 1, i.e. such that p > max(2/(α + 1); 2/(β + 1)). This readily follows by means of

p 2 eν|w| Γ(α + 1)Γ(β + 1) 2 (sν,α,β(w))p ≤ πp . m,n mp(α+1)/2np(β+1)/2 Thus we have proved the following

ν Proposition 4.3. Let α, β > 0. The singular values of Rw are given by (4.1). More- ν over, Rw is in p-Schatten class for every p such that p > max(2/(α + 1); 2/(β + 1)). ν Remark 4.4. Rw is not a trace class operator if α ≤ 1 or β ≤ 1. However, it is always a Hilbert–Schmidt operator for α, β > 0.

5 Connection to generalized Fractional Hankel transform ν The transform ψ 7−→ Rwψ(u, v), for fixed u, v ∈ D, seen as function in the variable w, is exactly the non trivial 2d fractional Fourier transform considered by Zayed in [20], to wit Z ν ν −ν|ζ|2 ν Fu,vψ(ξ) = ψ(ζ)Ku,v(ζ; ξ)e dλ(ζ) = Rξ ψ(u, v), C ν ν where Ku,v(ζ; ξ) is as in (1.3). The eigenfunctions of Fu,v are the Itô–Hermite polynomials, ν ν m n ν Fu,vψm,n = u v ψm,n. 2 The connection of fractional Fourier coefficients of given f ∈ Lν(C) to its fractional ν Fourier transform Fu,vf is given in Theorem 1.3 by means of the fractional Hankel transform in (1.7) which is a specific generalization of the classical Hankel transform []. ν,α To this end, we begin by interpreting Hu,v for integer order α = k as the radial part of ν the 2d Fourier transform Fu,v of k-rotationally symmetric function.

ikθ Proposition 5.1. For every rotational ψk(ζ) = Ψ(|ζ|)e , we have ξ !k/2 F ν ψ (ξ) = Hν,α(Ψ)(|ξ|). u,v k ξ u,v

ν −ν|ζ|2 Proof. By expanding the kernel function Ku,v(ζ; ξ)e in power series as

2 2 ! ∞ m+n m n n m ν −ν|ζ|2 −ν(|ζ| + uv|ξ| ) X ν (uξ) (vξ) ζ ζ Ku,v(ζ; ξ)e = exp m+n . 1 − uv m,n=0 (1 − uv) m!n! 8 ABDELHADI BENAHMADI AND ALLAL GHANMI

ν and using polar coordinates we see that Fu,vψ(ξ) takes the form ∞ 2π ∞ m n m+n ù,v X Z Z (uξ) (vξ) (ù,vr) 2 2 ψ(reiθ)ei(n−m)θ e−ù,v(r +uv|ξ| )rdrdθ π m,n=0 0 0 m!n! where ù,v stands for ù,v := ν/(1 − uv). Therefore, for every rotational symmetric ikθ function ψk(ζ) = Ψ(|ζ|)e , it reduces further to

!k/2 ∞ 2m+k ! Z ∞ √ 2m+k 2 2 uξ X (−i) −ù,v(r +uv|ξ| ) 2ù,v rΨ(r) iù,v uv|ξ|r e dr. vξ 0 m=0 m!(k + m)! Hence, by means of [1, p.222] ∞ !2n+α X 1 ξ Iα(ξ) := (5.1) n=0 n!Γ(α + n + 1) 2 α and Iα(−ξ) = (−1) Iα(ξ), it follows !k/2 √ ! Z ∞ 2 2 ν uξ 2ν uv −ù,v(r +uv|ξ| ) Fu,vψk(ξ) = 2ù,v rΨ(r)Ik |ξ|r e dr. vξ 0 1 − uv 2 Proof of Theorem 1.3. Notice first that for arbitrary f ∈ Lν(C), we have iθ X inθ f(re ) = gk(r)e . k∈Z Therefore, by setting ξ = ρeiϕ and making appeal of Proposition 5.1 we get ξ !k/2 F ν f(ξ) = X Hν,α(g )(ρ) = X Hν,α(g )(ρ)eikϕ. u,v ξ u,v k u,v k k∈Z k∈Z ν iϕ P ikϕ Accordingly, by identification to Fu,vf(ρe ) = Gk(ρ)e , we see that the Fourier k∈Z ν iθ iθ coefficients Gk(ρ) and gk(r) of θ 7−→ Fu,vf(ρe ) and θ 7−→ f(r ), respectively, satisfy ν,α Gk(ρ) = Hu,v (gk)(ρ).

References [1] Andrews G.E., Askey R., Roy R. Special functions. Encyclopedia of Mathematics and its Appli- cations, 71. Cambridge University Press: Cambridge; 1999. [2] Bargmann V., On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math., 14 (1961) 187–214. [3] Condon E.U., Immersion of the Fourier transform in a continuous group of functional transformations. Proc. Natl. Acad. Sci., USA 23 (1937) 158–164. [4] Chen Y., Liu Y., On the eigenfunctions of the complex Ornstein–Uhlenbeck operators. Kyoto J. Math., vol.54, no.3 (2014) 577–596 [5] Ghanmi A., Operational formulae for the complex Hermite polynomials Hp,q(z, z¯). Integral Trans- forms Spec. Funct., 2013; 24 (11):884-895. [6] Ghanmi A., Mehler’s formulas for the univariate complex Hermite polynomials and applications. Math. Methods Appl. Sci., 40, no. 18 (2017), 7540–7545. [7] Ghanmi A., On dual transform of fractional Hankel transform. Preprint 2020. [8] Hörmander L., Symplectic classiﬁcation of quadratic forms, and general Mehler formulas. Mathe- matische Zeitschrift, 219 (1995) 413-449. 2 [9] Intissar A., Intissar A. Spectral properties of the Cauchy transform on L2(C; e−|z| dλ). J. Math. Anal. Appl., (2006); 313 (2):400-418. DUAL OF 2D FRFT ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS 9

[10] Ismail M.E.H., Analytic properties of complex Hermite polynomials. Trans. Amer. Math. Soc., 2016; 368 (2):1189-1210. [11] Itô K., Complex multiple Wiener integral. Jap. J. Math., 1952; 22 :63-86. [12] Kibble W.F., An extension of a theorem of Mehler’s on Hermite polynomials. Proc. Cambridge Philos. Soc., 41 (1945) 12-15, [13] Louck J. D., Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods. Advances in Applied Mathematics, 2 (3) (1981) 239-249. [14] Mehler F.G., Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung. J. Reine Angew. Math., 1866; 66:161–176. [15] Slepian D., On the symmetrized Kronecker power of a matrix and extensions of Mehler’s formula for Hermite polynomials. SIAM Journal on Mathematical Analysis, 3 (4) (1972) 606-616, [16] Stanton D., Orthogonal polynomials and combinatorics, in Special Functions 2000: Current Per- spective and future directions, Kluwer, (2000) 389-410. [17] Srivastava H.M., Singhal J. P., Some extensions of the Mehler formula. Proc. Amer. Math. Soc., 31 (1972) 135-141. [18] Wiener N., Hermitian Polynomials and Fourier Analysis. Journal of Mathematics and Physics 8 (1929) 70-73. [19] Wünsche A., Transformations of Laguerre 2D-polynomials and their applications to quasiproba- bilities, J. Phys. A., 1999; 21:3179–3199. [20] Zayed A., Two-dimensional fractional Fourier transform and some of its properties. Integral Transforms Spec. Funct., 29 (7) (2018) :553–570. E-mail address: [email protected]

E-mail address: [email protected]

Analysis, P.D.E. & Spectral Geometry, Lab M.I.A.-S.I., CeReMAR, Department of Mathematics, P.O. Box 1014, Faculty of Sciences, Mohammed V University in Rabat, Morocco