The orthogonal complement of the Hilbert space associated to holomorphic Hermite polynomials

TL;DR

This paper investigates the orthogonal complement of a specific Hilbert space linked to holomorphic Hermite polynomials, providing explicit bases, kernels, and integral transforms to deepen understanding of its structure.

Contribution

It introduces a polyanalytic orthonormal basis and explicit formulas for reproducing kernels and Segal--Bargmann transforms for the space.

Findings

Explicit polyanalytic orthonormal basis constructed

Reproducing kernel functions derived

Segal--Bargmann integral transforms formulated

Abstract

We study the orthogonal complement of the Hilbert subspace considered by by van Eijndhoven and Meyers in [J. Math. Anal. Appl. 146 (1990), no. 1, 89--98} and associated to holomorphic Hermite polynomials. A polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal--Bargmann integral transforms are provided.

Equations

$$H_m(x)=(-1{)}^m{e}^{x^2}{\partial }_x^m{e}^{-x^2}=m!\sum _{k=0}^{[m/2]}\frac{(-1{)}^k}{k!}\frac{(2x{)}^{m-2k}}{(m-2k)!}.$$

$${\psi }_m^s(z)={\left(\frac{1-s}{\pi \sqrt{s}}\right)}^{1/2}{\left(\frac{1-s}{1+s}\right)}^{m/2}\frac{e^{-\frac{z^2}{2}}}{\sqrt{2^{m}m!}}{H}_m(z),$$

$${\int }_{\mathbb{C}}{\psi }_n^s(z)\overline{{\psi }_m^s(z)}{e}^{-\frac{1-s^2}{2s}|z{|}^2}{e}^{\frac{1+s^2}{4s}(z^2+{\overline{z}}^2)}d\lambda (z)={\delta }_{n,m},$$

$${\omega }_s(z,\overline{z})=e^{\frac{1+s^2}{4s}(z^2+{\overline{z}}^2)-\frac{1-s^2}{2s}|z{|}^2}.$$

$$[M_{\alpha }f](z):=M_{\alpha }(z)f(z)=e^{\frac{1+s^2}{4s}{z}^2}f(z),M_{\alpha }(z):=e^{\frac{1+s^2}{4s}{z}^2},$$

$${\tilde{\psi }}_m^s(z)=[M_{\alpha }{\psi }_m^s](z)$$

$$K^s(z,w)=\frac{1-s^2}{2\pi s}{e}^{-\frac{1+s^2}{4s}(z^2+{\overline{w}}^2)+\frac{1-s^2}{2s}z\overline{w}}.$$

$${\varphi }_m^s(z)=\frac{1}{\sqrt{\pi m!}}{\left(\frac{1-s^2}{2s}\right)}^{(m+1)/2}{e}^{-\frac{1+s^2}{4s}{z}^2}{z}^m$$

$$K^{{\mathcal{H}}^{\prime }}(z,w)=e^{\psi (z)}{K}^{\mathcal{H}}(z,w)e^{\overline{\psi (w)}}.$$

$$[{\mathcal{B}}_{s}f](z):={\int }_{\mathbb{R}}{B}_s(t,z)f(t)dt$$

$$B_s(t,z):={\left(\frac{1-s^2}{2\pi s\sqrt{s\pi }}\right)}^{1/2}\exp \left(-\frac{1}{2s}{t}^2-\frac{1}{2s}{z}^2+\frac{\sqrt{1-s^2}}{s}tz\right).$$

$$B_s(t,z):=\sum _{m=0}^{\infty }{f}_m(t){\psi }_m^s(z),$$

$$f_m(t)=\frac{e^{-\frac{t^2}{2}}}{\sqrt{2^{m}m!\sqrt{\pi }}}{H}_m(t)$$

$$\sum _{m=0}^{\infty }{f}_m(t){\psi }_m^s(z)$$

$$\sum _{m=0}^{\infty }\frac{{\lambda }^m}{2^{m}m!}{H}_m(t)H_m(z)=\frac{1}{\sqrt{1-{\lambda }^2}}\exp \left(\frac{-{\lambda }^2(t^2+z^2)+2\lambda tz}{1-{\lambda }^2}\right)$$

$$[{\mathcal{B}}_s^{-1}\phi ](t)={\int }_{\mathbb{C}}\phi (z)B_s(t,\overline{z}){\omega }_s(z,\overline{z})d\lambda (z).$$

$$\widetilde{B_s}(t,z)=\sum _{m=0}^{\infty }{f}_m(t){\varphi }_m^s(z).$$

$${\mathcal{F}}_n^{2,\nu }(\mathbb{C})=Ker{|}_{{\mathcal{H}}^{2,s}(\mathbb{C})}\left({\Delta }_{\nu }-n\nu Id\right)$$

$${\mathcal{H}}^{2,s}(\mathbb{C})=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{X}}_{n,s}(\mathbb{C}).$$

$$H_{m,n}^{\nu }(z,\overline{z}):=(-1{)}^{m+n}{e}^{\nu |z{|}^2}{\partial }_{\overline{z}}^m{\partial }_z^n\left(e^{-\nu |z{|}^2}\right)$$

$$[{\mathcal{W}}_n^{s}f](z,\overline{z})=\left(\frac{\nu }{\pi }\right){\left(\frac{{\nu }^n}{n!}\right)}^{1/2}{e}^{-\alpha z^2}{\int }_{\mathbb{C}}{e}^{-\nu |\xi {|}^2+\alpha {\xi }^2+\nu \overline{\xi }z}(\overline{z}-\overline{\xi }{)}^n\psi (\xi )d\lambda (\xi ).$$

$${\psi }_{m,n}^s(z,\overline{z})={\left(\frac{1-s}{\pi {\nu }^{n}n!\sqrt{s}}\right)}^{1/2}{\left(\frac{1-s}{1+s}\right)}^{m/2}\frac{e^{-\frac{z^2}{2}}}{\sqrt{2^{m}m!}}\left({\nabla }_{\nu ,\alpha -\frac{1}{2}}^n{H}_m\right)(z),$$

$$[{\mathcal{T}}_{k,n}^{\nu }\psi ](z,\overline{z})=\left(\frac{(-1{)}^n\nu }{\pi \sqrt{k!n!{\nu }^{k+n}}}\right){\int }_{\mathbb{C}}{e}^{-\nu |\xi {|}^2+\nu \overline{\xi }z}{H}_{k,n}^{\nu }(\xi -z,\overline{\xi }-\overline{z})\psi (\xi )d\lambda (\xi ),$$

$$[{\mathcal{T}}_{0,n}^{\nu }\psi ](z,\overline{z})={\left(\frac{1}{{\nu }^{n}n!}\right)}^{1/2}{\nabla }_{\nu ,0}^n\psi ,$$

$${\psi }_{m,n}^s(z,\overline{z})$$

$$={\left(\frac{1}{{\nu }^{n}n!}\right)}^{1/2}{M}_{-\alpha }{\nabla }_{\nu ,0}^n\left(M_{\alpha }{\psi }_m^s\right)(z)$$

$$={\left(\frac{1}{{\nu }^{n}n!}\right)}^{1/2}{\nabla }_{\nu ,\alpha }^n\left({\psi }_m^s\right)(z)$$

$$={\left(\frac{1}{{\nu }^{n}n!}\right)}^{1/2}{e}^{\frac{-z^2}{2}}{\nabla }_{\nu ,\alpha -\frac{1}{2}}^n\left(e^{\frac{z^2}{2}}{\psi }_m^s\right)(z),$$

$$[{\mathcal{W}}_n^s{]}^{-1}\psi (z)=\left(\frac{\nu }{\pi }\right){\left(\frac{{\nu }^n}{n!}\right)}^{1/2}{e}^{\alpha z^2}{\int }_{\mathbb{C}}{e}^{-\nu |\xi {|}^2-\alpha {\xi }^2+\nu \overline{\xi }z}(\xi -z{)}^n\psi (\xi )d\lambda (\xi ).$$

$$H_{m,n}^{\prime }(x,y;z,w|\tau )=m!n!\sum _{k=0}^{min(n,m)}\frac{(-\tau {)}^k}{k!}\frac{H_{n-k}^{\prime }(x,y)}{(n-k)!}\frac{H_{m-k}^{\prime }(z,w)}{(m-k)!}$$

$$K_n^s(z,w)=\left(\frac{1-s^2}{2\pi s}\right)\frac{(-1{)}^n}{n!{\nu }^n}{e}^{\nu z\overline{w}-\alpha (z^2+{\overline{w}}^2)}{H}_{n,n}^{\nu }(z-w,\overline{z}-\overline{w}).$$

$$K_n^{{\mathcal{F}}^{2,\nu }(\mathbb{C}{)}_n}(z,w)=\left(\frac{\nu }{\pi }\right)\frac{(-1{)}^n}{n!{\nu }^n}{e}^{\nu z\overline{w}}{H}_{n,n}^{\nu }(z-w,\overline{z}-\overline{w}).$$

Full text

1. Introduction

The known Hermite polynomials and their different generalizations have been one of the most interesting fields for research, since their introduction by Lagrange and Chebyshev. They appear in a wide spectrum of research domains including enginery, pure and applied mathematics, and different branches of physics. The classical ones on the real line $\mathbb{R}$ are defined by ([10, 11, 12])

[TABLE]

Here and elsewhere after, we use $\partial_{x}$ to denote the partial differential operator $\partial/\partial_{x}$. They can be extended to the whole complex plane $\mathbb{C}$ by replacing the real $x$ by the complex variable $z$, leading to the class of holomorphic Hermite polynomials $H_{m}(z)$. The last ones inherit the most of the algebraic properties of $H_{m}(x)$ by analytic continuation. Moreover, they possess further interesting analytic properties. The associated functions

[TABLE]

for given fixed $0<s<1$, satisfy the orthogonal property ([13])

[TABLE]

where $d\lambda(z)=dxdy$ being the Lebesgue measure on $\mathbb{C}\equiv\mathbb{R}^{2}$. This is to say that the functions $\psi^{s}_{n}(z)$ in (1.1) form an orthonormal system in the Hilbert space $\mathscr{H}^{2,s}(\mathbb{C}):=L^{2}(\mathbb{C},\omega_{s}d\lambda)$, where the weight function $\omega_{s}$ is given by

[TABLE]

Equivalently, if $M_{\alpha}$ denotes the multiplication operator

[TABLE]

with $\alpha=\alpha_{s}=\frac{1+s^{2}}{4s}$, then the functions

[TABLE]

form an orthonormal system in $\mathcal{L}^{2,\nu}(\mathbb{C}):=L^{2}(\mathbb{C},e^{-\nu|z|^{2}}d\lambda)$, where $\nu={\nu_{s}}=\frac{1-s^{2}}{2s}$. Accordingly, we define the Hilbert subspace $\mathcal{X}_{s}(\mathbb{C})$ as in [13] by $\mathcal{X}_{s}(\mathbb{C})=\mathcal{H}ol\cap\mathscr{H}^{2,s}(\mathbb{C})$. Its companion $\mathcal{F}^{2,\nu}(\mathbb{C})=\mathcal{H}ol\cap\mathcal{L}^{2,\nu}(\mathbb{C})=M_{\alpha}(\mathcal{X}_{s}(\mathbb{C}))$ is the classical Bargmann–Fock space of weight ${\nu}$ (see e.g. [1, 4]).

The aim of the present paper is three folds

1. 1

Review and complete the study of the space $\mathcal{X}_{s}(\mathbb{C})$. In particular, we provide the associated Segal–Bargmann transforms for the configuration space $L^{2}(\mathbb{R})$. See Section 2. 2. 2

Study a Hilbertian decomposition of $\mathscr{H}^{2,s}(\mathbb{C})$ in terms of some reproducing kernel Hilbert subspaces $\mathcal{X}_{n,s}(\mathbb{C})$, and provide to each $\mathcal{X}_{n,s}(\mathbb{C})$ an orthonormal basis generalizing the ones in (1.4) to the polyanalytic setting, as well as the explicit expression of the reproducing kernel of $\mathcal{X}_{n,s}(\mathbb{C})$. See Section 3. 3. 3

We also give the corresponding Segal–Bargmann integral transform. See Section 3.

2. Complements on $\mathcal{X}_{s}(\mathbb{C})$

We begin with the following

Proposition 2.1** ([13]).**

The functions $\psi^{s}_{n}$ constitute an orthonormal basis of the reproducing kernel Hilbert space $\mathcal{X}_{s}(\mathbb{C})$ with kernel given explicitly by

[TABLE]

Proof.

The proof of (2.1) can be handled by invoking the unitary operator $M_{\alpha}$ in (1.3) and observing that the functions

[TABLE]

form an orthonormal basis of $\mathcal{X}_{s}(\mathbb{C})$, so that one concludes for the explicit expression of $K^{s}(z,w)$ by performing $K^{s}(z,w)=\sum\limits_{m=0}^{+\infty}\phi^{s}_{m}(z)\overline{\phi^{s}_{m}(w)}$ and next using the generating function of the Hermite polynomials $H_{n}(z)$ ([10, p. 130]). ∎

Remark 2.2**.**

The expression of the reproducing kernel can also be proved in an easy way by making appeal to the following general principle. Let $\mathcal{H}$ be a separable reproducing kernel Hilbert space (RKHS) on the complex plane and denotes by $K^{\mathcal{H}}$ its reproducing kernel function. If $M$ is a multiplication operator by a function $M(z):=e^{\psi(z)}$. Then, $\mathcal{H}^{\prime}=M\mathcal{H}$ is a RKHS whose kernel function is given by

[TABLE]

Remark 2.3**.**

The space $\mathop{\cup}\limits_{0<s<1}\mathcal{X}_{s}(\mathbb{C})=S^{1/2}_{1/2}$ is the Gelfand–Shilov space (of holomorphic functions) extended to $\mathbb{C}$ (see [13]).

In the sequel, we consider the integral transform of Segal–Bargmann type

[TABLE]

associated to the kernel function

[TABLE]

Then, we assert

Theorem 2.4**.**

The transform $\mathscr{B}_{s}$ defines a unitary isometric integral transform from the configuration Hilbert space $L^{2}(\mathbb{R})$ onto $\mathcal{X}_{s}(\mathbb{C})$.

Proof.

The kernel function $B_{s}(t,z)$ in (2) can be rewritten as

[TABLE]

where

[TABLE]

is an orthonormal basis of $L^{2}(\mathbb{R})$. Indeed, we have

[TABLE]

The rest of the proof is straightforward making use of the Mehler formula for the Hermite polynomials extended to the complex plane, to wit ([9, p.174, Eq. (18)], see also [10, p.198, Eq. (2)])

[TABLE]

valid for every fixed $0<\lambda<1$. ∎

Remark 2.5**.**

By means of (2.4) and (2.6), we have $[\mathscr{B}_{s}f_{m}](z)=\psi^{s}_{m}(z).$ Moreover, the inversion formula of $\mathscr{B}_{s}$ is given by

[TABLE]

Remark 2.6**.**

By considering $\widetilde{B_{s}}(t,z):=s^{1/4}B_{s}(s^{1/2}t,z)$, we define an integral transform $\widetilde{\mathscr{B}}_{s}$ from $L^{2}(\mathbb{R})$ onto $\mathcal{X}_{s}(\mathbb{C})$ such that $[\widetilde{\mathscr{B}}_{s}f_{m}n](z)=\phi^{s}_{m}(z)$, where $\phi^{s}_{m}$ are as in (2.2), since

[TABLE]

3. A special orthonormal basis of $\mathscr{H}^{2,s}(\mathbb{C})$

The multiplication operator $M_{\alpha}:f\longmapsto M_{\alpha}f=e^{\alpha z^{2}}f$ defines a unitary operator from $\mathscr{H}^{2,s}(\mathbb{C})$ onto $\mathcal{L}^{2,\nu}(\mathbb{C})$. Moreover, it maps isometrically the Hilbert subspace $\mathcal{X}_{s}(\mathbb{C})$ onto the Bargmann–Fock space $\mathcal{F}^{2,\nu}(\mathbb{C})$. Therefore, an orthogonal decomposition of $\mathscr{H}^{2,s}(\mathbb{C})$ can be deduced easily from the one of $\mathcal{L}^{2,\nu}(\mathbb{C})$, $\mathcal{L}^{2,\nu}(\mathbb{C})=\bigoplus_{n=0}^{\infty}\mathcal{F}^{2,\nu}_{n}(\mathbb{C}),$ given in terms of the polyanalytic Hilbert spaces

[TABLE]

where $\Delta_{\nu}:=-\partial_{z}\partial_{\overline{z}}+\nu\overline{z}\partial_{\overline{z}}$ and with $\mathcal{F}^{2,\nu}_{0}(\mathbb{C})=\mathcal{F}^{2,\nu}(\mathbb{C})$. See for e.g. [7] for details. In fact, the consideration of $\mathcal{X}_{n,s}(\mathbb{C}):=M_{-\alpha}\mathcal{F}^{2,\nu}_{n}(\mathbb{C})$ leads to the orthogonal decomposition

[TABLE]

An immediate orthonormal basis of $\mathcal{X}_{n,s}(\mathbb{C})$ is then given by $e^{-\alpha z^{2}}H_{m,n}^{{\nu}}(z,\overline{z})$ for varying $m,n=0,1,2,\cdots$, where

[TABLE]

denotes the weighted polyanalytic complex Hermite polynomials [5, 6, 8], generalizing the monomials ${\nu}^{m}z^{m}=H_{m,0}^{{\nu}}(z,\overline{z})$.

The main aim in this section is to provide another ”nontrivial” orthonormal basis $\psi^{s}_{m,n}(z,\overline{z})$ of $\mathscr{H}^{2,s}(\mathbb{C})$, consisting of polyanalytic functions generalizing $\psi^{s}_{m}$ and whose first elements are the holomorphic functions $\psi^{s}_{m}(z)$ in (1.4), i.e., $\psi^{s}_{m,0}(z,\overline{z})=\psi^{s}_{m}(z)$. and obtained an appropriate basis of the space $\mathcal{X}_{n,s}(\mathbb{C})$. The introduction of $\mathcal{X}_{n,s}(\mathbb{C})$ entails the consideration of the integral transform

[TABLE]

Then, we can prove the following

Theorem 3.1**.**

*The transform $\mathscr{W}^{s}_{n}$ is a unitary integral transform from $\mathcal{X}_{s}(\mathbb{C})$ onto $\mathcal{X}_{n,s}(\mathbb{C})$. Moreover, the functions *

[TABLE]

where $\nabla_{\nu,\alpha}:=-\partial_{z}+\nu\overline{z}-2\alpha z$, form an orthonormal basis of $\mathcal{X}_{n,s}(\mathbb{C})$.

Proof.

The proof lies essentially on the observation that the unitary operator $\mathscr{W}^{s}_{n}$ can be rewritten as $\mathscr{W}^{s}_{n}=M_{-\alpha}\mathscr{T}^{\nu}_{0,n}M_{\alpha}$, where $\mathscr{T}^{\nu}_{k,n}$ is the integral transform considered in [1, Eq. (2.17)] and given by

[TABLE]

as well as on the fact that $\psi^{s}_{m,n}(z,\overline{z}):=[\mathscr{W}^{s}_{n}\psi^{s}_{m}](z,\overline{z})$. Thus, by means of [1, Theorem 2.12], keeping in mind the fact that the polynomials $H^{\nu}_{m,n}(z,\overline{z})=:\nabla_{\nu,0}^{n}(z^{m})$ is an orthogonal basis of $\mathcal{L}^{2,\nu}(\mathbb{C})$ [8, 5], the following

[TABLE]

holds true for every nonnegative integers $n$ and any $\psi\in\mathcal{L}^{2,\nu}(\mathbb{C})\cap\mathcal{C}^{n}(\mathbb{C})$. The rest of the second assertion is straightforward since the functions $\psi^{s}_{m}$ form an orthonormal basis of $\mathcal{X}_{s}(\mathbb{C})$. The explicit expression of $\psi^{s}_{m,n}(z,\overline{z})$ follows by direct computation. Indeed, we have

[TABLE]

since $\nabla_{\nu,a}\left(M_{\gamma}\psi\right)=M_{\gamma}\nabla_{\nu,a+\gamma}\psi$ and $\nabla_{\nu,0}^{n}\left(M_{\gamma}\psi\right)=M_{\gamma}\nabla_{\nu,\gamma}^{n}\psi$. ∎

Remark 3.2**.**

The inverse of $\mathscr{W}^{s}_{n}:\mathcal{X}_{s}(\mathbb{C})\longrightarrow\mathcal{X}_{n,s}(\mathbb{C})$ is given by $[\mathscr{W}^{s}_{n}]^{-1}=M_{\alpha}\mathscr{T}^{\nu}_{n,0}M_{-\alpha}$, More explicitly

[TABLE]

Remark 3.3**.**

The new class of functions in (3.1) generalizes the one studied in [2] and the previous theorem provide an integral representation of the special functions $\psi^{s}_{m,n}(z,\overline{z})$. Moreover, it is closely connected to the polynomials

[TABLE]

*in [3], where $H^{\prime}_{n}(x,y):=i^{n}y^{\frac{n}{2}}H_{n}\left(\frac{x}{2i}y^{-\frac{1}{2}}\right).$ *

The considered space $\mathcal{X}_{n,s}(\mathbb{C})$ is a reproducing kernel Hilbert space for the point evaluation map in $\mathcal{X}_{n,s}(\mathbb{C})$ is continuous. This, can be recovered easily by means of Remark 2.2. Thus, we assert

Theorem 3.4**.**

The explicit expression of the reproducing kernel of $\mathcal{X}_{n,s}(\mathbb{C})$ is given by

[TABLE]

Proof.

By means of Remark 2.2, the reproducing kernel $K^{s}_{n}(z,w)$ of $\mathcal{X}_{n,s}(\mathbb{C})$ obeys (2.3). Hence, we have $K^{s}_{n}(z,w)=M_{\alpha}(z)K^{\mathcal{F}^{2,\nu}(\mathbb{C})}(z,w)\overline{M_{\alpha}(w)}$. where $K^{\mathcal{F}^{2,\nu}(\mathbb{C})_{n}}$ is the reproducing kernel of the generalized Bargmann space $\mathcal{F}^{2,\nu}(\mathbb{C})_{n}$ given by [7]

[TABLE]

∎

Remark 3.5**.**

For $n=0$ we recover the reproducing kernel of the Hilbert space $\mathcal{X}_{s}(\mathbb{C})$ in Proposition 2.1.

Remark 3.6**.**

*The identity *

[TABLE]

or equivalently

[TABLE]

*holds true by comparing the result of Theorem 3.4 to the fact that the reproducing kernel $K^{s}_{n}$ can be rewritten as $K^{s}_{n}(z,w)=\sum\limits_{m=0}^{+\infty}\psi^{s}_{m,n}(z)\overline{\psi^{s}_{m,n}(w)}$, for $\{\psi^{s}_{m,n}(z,\overline{z}),m=0,1,2,\cdots\}$, in (3.1), being an orthonormal basis of $\mathcal{X}_{n,s}(\mathbb{C})$. *

We conclude this section by giving the explicit expression of the generalized Segal–Bargmann integral transform for the spaces $\mathcal{X}_{n,s}(\mathbb{C})$. We have to consider the weighted configuration space $L^{2,\nu}(\mathbb{R})$ instead of $L^{2}(\mathbb{R})$, where $\nu>0$. It is the Hilbert space of all square integrable $\mathbb{C}$-valued functions on $\mathbb{R}$ with respect to the Gaussian measure $e^{-\nu x^{2}}dx$, for which the rescaled Hermite polynomials

[TABLE]

form an orthonormal basis. The associated coherent states transform from $L^{2,\nu}(\mathbb{R})$ onto $\mathcal{X}_{n,s}(\mathbb{C})$ mapping $g^{\nu}_{m}$ to $\psi^{s}_{m,n}$ is given by

[TABLE]

where the kernel function $S^{s}_{n}(x,z)$ is given by

[TABLE]

For fixed $a>0$, $b\in\mathbb{R}$ and $c\in\mathbb{C}$, we define $I_{n}^{a,b}(z,\overline{z}|c)$ to be the class of polyanalytic polynomials in [2],

[TABLE]

Theorem 3.7**.**

We have

[TABLE]

Moreover, the transform $\mathscr{S}^{s}_{n}$ defines an isomtric transform from $L^{2,\nu}(\mathbb{R})$ onto $\mathcal{X}_{n,s}(\mathbb{C})$.

Proof.

We need only to prove the closed formula (3.4) for $S^{s}_{n}(x,z)$. The rest holds true for general coherent state transformations on the reproducing kernel Hilbert spaces likes $\mathcal{X}_{n,s}(\mathbb{C})$. Indeed, starting from (3.3) and (3.1) and applying the Mehler formula (2.8) the expression of $S^{s}_{n}(x,z)$ reduces further to

[TABLE]

Using the fact $\nabla_{\nu,\gamma}f=-e^{\nu|z|^{2}-\gamma z^{2}}\partial_{z}\left(e^{-\nu|z|^{2}+\gamma z^{2}}f\right)$, we get

[TABLE]

by induction, and therefore

[TABLE]

Subsequently

[TABLE]

∎

4. Concluding remarks

In the previous section the space $\mathcal{X}_{n,s}(\mathbb{C})$ are realized as the image of $\mathcal{X}_{s}(\mathbb{C})$ the integral transform $\mathcal{W}^{s}_{n}$ or also as the image of $L^{2,\nu}(\mathbb{R})$ by the generalized Segal–Bargmann transform $\mathscr{S}^{s}_{n}$. Another realization of $\mathcal{X}_{n,s}(\mathbb{C})$ is by considering the $n$-th standard Segal–Bargmann transform [2]

[TABLE]

from $L^{2,\nu}(\mathbb{R})$ onto $\mathcal{F}^{2,\nu}_{n}(\mathbb{C})$. Indeed, one has to deal with $\mathscr{B}^{\prime}_{\nu,n}:L^{2,\nu}(\mathbb{R})\longrightarrow\mathcal{X}_{n,s}(\mathbb{C})$,

[TABLE]

It is clear that for every fixed $b$, the functions $[\mathscr{B}^{\prime}_{\nu,n}]^{-1}\psi_{m,n}$ form an orthonormal basis of $L^{2,\nu}(\mathbb{R})$. But, there is no clear evidence if they are the same or not. We claim that $\left([\mathscr{B}^{\prime}_{\nu,n}]^{-1}\psi_{m,n}\right)$ do not depend of $n$. The corresponding Poisson kernel can be given explicitly leading to a nontrivial $1d$-fractional Fourier transform for the Hilbert space $L^{2,\nu}(\mathbb{R})$.