Holomorphic Hermite functions and ellipses

TL;DR

This paper explores holomorphic Hermite functions linked to ellipses on the complex plane, connecting them to Segal-Bargmann spaces and extending prior work on these functions and their applications.

Contribution

It demonstrates that holomorphic Hermite functions associated with ellipses are characterized by specific ellipse cases and relate to Segal-Bargmann spaces via Bargmann-type transforms.

Findings

Holomorphic Hermite functions are determined by certain ellipses.

Reproducing kernel Hilbert spaces correspond to Segal-Bargmann spaces.

Connections established between ellipse-based functions and Bargmann transforms.

Abstract

In 1990 van Eijnghoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand-Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal-Bargmann spaces determined by the Bargmann-type transforms introduced by Sjoestrand.

Equations

$$\Vert \phi {\Vert }_s^2={\int }_{\mathbb{C}}|\phi (z){|}^2\exp \left(-\frac{1-s^2}{2s}|z{|}^2+\frac{1+s^2}{4s}(z^2+{\bar{z}}^2)\right)L(dz)<\infty ,$$

$$(\phi ,\psi {)}_s={\int }_{\mathbb{C}}\phi (z)\overline{\psi (z)}\exp \left(-\frac{1-s^2}{2s}|z{|}^2+\frac{1+s^2}{4s}(z^2+{\bar{z}}^2)\right)L(dz),\phi ,\psi \in {\mathcal{X}}_s(\mathbb{C}).$$

$${\psi }_n^s(z)=b_{nn}(s{)}^{-1/2}{e}^{-z^2/2}{H}_n(z),n=0,1,2,\dots ,$$

$$b_{mn}(s)=\frac{\pi \sqrt{s}}{1-s}{2}^n{\left(\frac{1+s}{1-s}\right)}^{n}n!{\delta }_{mn},m,n=0,1,2,\dots .$$

$$H_n(z)=(-1{)}^n{e}^{z^2}{\left(\frac{d}{dz}\right)}^n{e}^{-z^2},n=0,1,2,\dots .$$

$$\phi (z)={\int }_{\mathbb{C}}{K}_s(z,\zeta )\phi (\zeta )\exp \left(-\frac{1-s^2}{2s}|\zeta {|}^2+\frac{1+s^2}{4s}({\zeta }^2+{\bar{\zeta }}^2)\right)L(d\zeta )$$

$$K_s(z,\zeta )=\frac{1-s^2}{2\pi s}\exp \left(\frac{1-s^2}{2s}z\bar{\zeta }-\frac{1+s^2}{4s}(z^2+{\bar{\zeta }}^2)\right).$$

$$Bf(z)=2^{-1/2}{\pi }^{-3/4}{\int }_{\mathbb{R}}{e}^{-z^2/4+zx-x^2/2}f(x)dx,z\in \mathbb{C}.$$

$$|e^{-z^2/4+zx-x^2/2}|=\mathcal{O}(e^{-x^2/4})$$

$$(f,g{)}_{L^2}={\int }_{\mathbb{R}}f(x)\overline{g(x)}dx,f,g\in L^2(\mathbb{R}).$$

$$(\phi ,\psi {)}_{{\mathcal{H}}_B}={\int }_{\mathbb{C}}\phi (z)\overline{\psi (z)}{e}^{-|z{|}^2/2}L(dz),\phi ,\psi \in {\mathcal{H}}_B(\mathbb{C}).$$

$$B^{\ast }\phi (x)=2^{-1/2}{\pi }^{-3/4}{\int }_{\mathbb{C}}{e}^{-{\bar{z}}^2/4+\bar{z}x-x^2/2}\phi (z)e^{-|z{|}^2/2}L(dz),x\in \mathbb{R}.$$

$$B\circ B^{\ast }F(z)=\frac{1}{2\pi }{\int }_{\mathbb{C}}{e}^{z\bar{\zeta }/2}F(\zeta )e^{-|\zeta {|}^2/2}L(d\zeta ),F\in L_B^2(\mathbb{C}).$$

$$E_{\rho }(\alpha ,\beta )=\{x+i\xi \in \mathbb{C}|(x,\xi )\in {\mathbb{R}}^2,|\alpha x+i(\beta x+\xi )|⩽\rho \}.$$

$$\zeta =\alpha x+i(\beta x+\xi )=\frac{\alpha +1+i\beta }{2}z+\frac{\alpha -1+i\beta }{2}\bar{z},$$

$${\Psi }_0^{\alpha ,\beta }(z)=\exp \left(\frac{\mu }{4}{z}^2\right),\mu =\mu (\alpha ,\beta )=\frac{1-{\alpha }^2-{\beta }^2+2i\beta }{1+{\alpha }^2+{\beta }^2},$$

$${\Psi }_n^{\alpha ,\beta }(z)=\left\{e^{\lambda z^2/2}{\left(\frac{d}{dz}\right)}^n{e}^{-\lambda z^2/2}\right\}{\Psi }_0^{\alpha ,\beta }(z),n=1,2,3,\dots ,$$

$$\lambda =\lambda (\alpha ,\beta )=\frac{2{\alpha }^2}{(1+{\alpha }^2+{\beta }^2)(1-{\alpha }^2-{\beta }^2-2i\beta )}.$$

$$|{\Psi }_0^{\alpha ,\beta }(z){|}^2{e}^{-|z{|}^2/2}=\exp \left(-\frac{|\zeta {|}^2}{1+{\alpha }^2+{\beta }^2}\right),$$

$$Tf(z)=C_{\varphi }{\int }_{\mathbb{R}}{e}^{i\varphi (z,x)}f(x)dx,z\in \mathbb{C},$$

$$\varphi (z,x)=\frac{a}{2}{z}^2+bzx+\frac{c}{2}{x}^2$$

$$\Phi (z)$$

$$\Psi (z,\zeta )$$

$$(\phi ,\psi {)}_{{\mathcal{H}}_{\Phi }}={\int }_{\mathbb{C}}\phi (z)\overline{\psi (z)}{e}^{-2\Phi (z)}L(dz),\phi ,\psi \in {\mathcal{H}}_{\Phi }(\mathbb{C}).$$

$$T^{\ast }\phi (x)=C_{\varphi }{\int }_{\mathbb{C}}{e}^{-i\overline{\varphi (z,x)}}\phi (z)e^{-2\Phi (z)}L(dz),x\in \mathbb{R}.$$

$$T\circ T^{\ast }F(z)=C_{\Phi }{\int }_{\mathbb{C}}{e}^{2\Psi (z,\bar{\zeta })}F(\zeta )e^{-2\Phi (\zeta )}L(d\zeta ),F\in L_{\Phi }^2(\mathbb{C}),$$

$${\mathcal{X}}_s(\mathbb{C})\ni \phi (z)\mapsto \sqrt{\frac{s}{1-s^2}}\phi \left(\frac{s}{1-s^2}z\right)\exp \left(\frac{1}{4}\frac{1+s^2}{1-s^2}{z}^2\right)\in {\mathcal{H}}_B(\mathbb{C})$$

$${\mathcal{H}}_B(\mathbb{C})\ni \psi (z)\mapsto \sqrt{\frac{1-s^2}{s}}\psi \left(\frac{1-s^2}{s}z\right)\exp \left(-\frac{1+s^2}{4s}{z}^2\right)\in {\mathcal{X}}_s(\mathbb{C}),$$

$${\psi }_n^s(z)\times \exp \left(-\frac{1-s^2}{4s}|z{|}^2+\frac{1+s^2}{4s}{z}^2\right)$$

$$=$$

$$\partial E_{\rho }(\sqrt{s},0)=\{x+i\xi \in \mathbb{C}|(x,\xi )\in {\mathbb{R}}^2,s{x}^2+{\xi }^2={\rho }^2\}.$$

Full text

Supported by the JSPS Grant-in-Aid for Scientific Research #16K05221.

1. Introduction

Let $0<s<1$. We denote by $\mathscr{X}_{s}(\mathbb{C})$ the set of all holomorphic functions on $\mathbb{C}$ satisfying an integrability condition

[TABLE]

where $L(dz)$ is the Lebesgue measure on $\mathbb{C}\simeq\mathbb{R}^{2}$. The function space $\mathscr{X}_{s}(\mathbb{C})$ is a Hilbert space equipped with an inner product

[TABLE]

The quantity $\lVert\cdot\rVert_{s}$ coincides with the norm induced by the inner product. In [1] van Eijndhoven and Meyers first introduced the function space $\mathscr{X}_{s}(\mathbb{C})$ and holomorphic Hermite functions $\psi^{s}_{n}$ defined by

[TABLE]

[TABLE]

[TABLE]

They proved the following properties.

Theorem 1** (van Eijndhoven and Meyers [1]).**

• •

$\mathscr{X}_{s}(\mathbb{C})$* is a reproducing kernel Hilbert space with a reproducing formula*

[TABLE]

for $z\in\mathbb{C}$ and $\phi\in\mathscr{X}_{s}(\mathbb{C})$, where the integral kernel is given by

[TABLE]

• •

$\{\psi^{s}_{n}\}_{n=0}^{\infty}$* is a complete orthonormal system of $\mathscr{X}_{s}(\mathbb{C})$.*

• •

$\displaystyle\bigcup_{0<s<1}\mathscr{X}_{s}(\mathbb{C})=S^{1/2}_{1/2}$*, where $S^{1/2}_{1/2}$ is a Gelfand-Shilov space extended on $\mathbb{C}$ *(See [2]).

After that, the function space $\mathscr{X}_{s}(\mathbb{C})$ and the system of holomorphic Hermite functions $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ have been applied to studying quantization on $\mathbb{C}$ and related problems (see [3, 4, 5]), combinatorics and counting (see [6]) and etc.

Most recently, the author happened to introduce holomorphic Hermite functions associated with ellipses on the complex plane in [7]. In order to explain this, we here introduce the standard Bargmann transform, the standard Segal-Bargmann space and their basic properties quickly. See [7, 8] for the detail. For any rapidly decreasing function $f(x)$ on the real line $\mathbb{R}$, its standard Bargmann transform $Bf(z)$ is defined by

[TABLE]

The Bargmann transform can be extended for all the tempered distributions on $\mathbb{R}$ since

[TABLE]

for any fixed $z\in\mathbb{C}$. We denote the set of all square integrable functions on $\mathbb{R}$ by $L^{2}(\mathbb{R})$, which is a Hilbert space equipped with an inner product

[TABLE]

Set $\lVert{f}\rVert_{L^{2}}=\sqrt{(f,f)_{L^{2}}}$ for short. Moreover, we denote the set of all square integrable holomorphic functions on $\mathbb{C}$ with respect to a weighted measure $e^{-\lvert{z}\rvert^{2}/2}L(dz)$ by $\mathscr{H}_{B}(\mathbb{C})$, which is said to be the standard Segal-Bargmann space on $\mathbb{C}$ and is a Hilbert space equipped with an inner product

[TABLE]

Set $\lVert{\varphi}\rVert_{\mathscr{H}_{B}}=\sqrt{(\varphi,\varphi)_{\mathscr{H}_{B}}}$ for short. It is well-known that the Bargmann transform $B$ is a Hilbert space isomorphism of $L^{2}(\mathbb{R})$ onto $\mathscr{H}_{B}(\mathbb{C})$, that is, $B$ is bijective and $(Bf,Bg)_{\mathscr{H}_{B}}=(f,g)_{L^{2}}$ for $f,g\in L^{2}(\mathbb{R})$. The inverse of $B$ is given by the adjoint of $B$, that is,

[TABLE]

Note that $B^{\ast}$ can be extended for $L^{2}_{B}(\mathbb{C})$, which is the set of all square integrable functions on $\mathbb{C}$ with respect to a weighted measure $e^{-\lvert{z}\rvert^{2}/2}L(dz)$. This is also a Hilbert space. The definition of its inner product and norm is the same as those of $\mathscr{H}_{B}(\mathbb{C})$. Then $\mathscr{H}_{B}(\mathbb{C})$ is a closed subspace of $L^{2}_{B}(\mathbb{C})$, and the projection of $L^{2}_{B}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$ is given by

[TABLE]

In particular, $\varphi=B{\circ}B^{\ast}\varphi$ for $\varphi\in\mathscr{H}_{B}(\mathbb{C})$.

Suppose that $\alpha>0$ and $\beta\in\mathbb{R}$ are fixed constants satisfying $(\alpha,\beta)\neq(1,0)$. For $\rho>0$, we consider an elliptic disk of the form

[TABLE]

Note that $\partial E_{\rho}(\alpha,\beta)$ is an ellipse whose major and minor axes join at the origin of $\mathbb{C}$. For $z=x+i\xi$, set

[TABLE]

[TABLE]

[TABLE]

[TABLE]

More precisely, $\{\Psi^{\alpha,\beta}\}_{n=1}^{\infty}$ is generated by the creation operators, which is the adjoint of the annihilation operator for $\Psi^{\alpha,\beta}_{0}$. Then we have

[TABLE]

which shows that the function $\Psi^{\alpha,\beta}_{0}$ and the system $\{\Psi^{\alpha,\beta}_{n}\}_{n=1}^{\infty}$ are concerned with the elliptic disk $E_{\rho}(\alpha,\beta)$ in some sense. One of the results in [7] is the following.

Theorem 2** ([7, Theorem 4.2]).**

The family $\{\Psi^{\alpha,\beta}_{n}\}_{n=0}^{\infty}$ is a complete orthogonal system of $\mathscr{H}_{B}(\mathbb{C})$.

One of the purposes of the present paper is to understand Theorem 1 and its subsequent results in the framework of the standard Segal-Bargmann space $\mathscr{H}_{B}(\mathbb{C})$.

The standard Bargmann transform $B$, the standard Segal-Bargmann space $\mathscr{H}_{B}(\mathbb{C})$ and related objects are generalised. In fact, Sjöstrand constructed more general framework and applied it to studying microlocal analysis. See, e.g., [9, 10] and references therein. In what follows we recall Sjöstrand’s theory restricted on $\mathbb{R}$ quickly. The generalisation of the Bargmann transform $B$ is given as a global Fourier integral operator of the form

[TABLE]

where $\phi(z,x)$ is a complex-valued quadratic phase function of the form

[TABLE]

for an appropriate function $f$ with assumptions $b\neq 0$ and $\operatorname{Im}c>0$, and $C_{\phi}=2^{-1/2}\pi^{-3/4}\lvert{b}\rvert(\operatorname{Im}c)^{-1/4}$. We call $T$ a Bargmann-type transform. Note that $T$ can be also extended for all the tempered distributions on $\mathbb{R}$ since $\lvert{e^{i\phi(z,x)}}\rvert=\mathcal{O}(e^{-(\operatorname{Im}c)x^{2}/4})$. The case of $a=i/2$, $b=-i$ and $c=i$ corresponds to the standard Bargmann transform $B$. Set

[TABLE]

We denote the set of all square integrable holomorphic functions on $\mathbb{C}$ with respect to a weighted measure $e^{-2\Phi(z)}L(dz)$ by $\mathscr{H}_{\Phi}(\mathbb{C})$, which is a Hilbert space equipped with an inner product

[TABLE]

Set $\lVert\varphi\rVert_{\mathscr{H}_{\Phi}}=\sqrt{(\varphi,\varphi)_{\mathscr{H}_{\Phi}}}$ for short. Similarly we define $L^{2}_{\Phi}(\mathbb{C})$ which is a Hilbert space consisting of all square integrable functions on $\mathbb{C}$ with respect to the weighted measure $e^{-2\Phi(z)}L(dz)$. We denote its inner product and norm by $(\cdot,\cdot)_{L^{2}_{\Phi}}$ and $\lVert\cdot\rVert_{L^{2}_{\Phi}}$ respectively. The operator $T$ gives a Hilbert space isomorphism of $L^{2}(\mathbb{R})$ onto $\mathscr{H}_{\Phi}(\mathbb{C})$, that is, $T$ is bijective and $(Tf,Tg)_{\mathscr{H}_{\Phi}}=(f,g)_{L^{2}}$ for $f,g\in L^{2}(\mathbb{R})$. The inverse of $T$ is the formal adjoint $T^{\ast}$ which is concretely given by

[TABLE]

Moreover, $\mathscr{H}_{\Phi}(\mathbb{C})$ is a closed subspace of $L^{2}_{\Phi}(\mathbb{C})$, and the projection operator is given by

[TABLE]

where $C_{\Phi}=\lvert{b}\rvert/2\pi\operatorname{Im}c$. In particular, $\varphi=T{\circ}T^{\ast}\varphi$ for $\varphi\in\mathscr{H}_{\Phi}(\mathbb{C})$.

The purpose of the present paper is to understand the function space $\mathscr{X}_{s}(\mathbb{C})$ and the family of holomorphic Hermite functions $\{\psi^{s}_{n}\}_{n=0}^{\infty}$, and the subsequent results in the framework of Sjöstrand’s microlocal analysis. More precisely, in Section 2 we first study the properties of $\mathscr{X}_{s}(\mathbb{C})$ and $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ in the framework of the standard Bargmann transform. In particular, we shall understand $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ from a view point of ellipses originated in [7]. Finally, in Section 3 we study $\mathscr{X}_{s}(\mathbb{C})$ and $\{\psi^{s}_{n}\}_{n=0}^{\infty}$, and some of subsequent results in the framework of Sjöstrand.

2. $\mathscr{X}_{s}(\mathbb{C})$ and ellipses

In this section we shall understand Theorem 1 from a view point of [7, Section 4]. Our results in the present section are the following.

Theorem 3**.**

$\mathscr{X}_{s}(\mathbb{C})$* and $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ are essentially determined by the ellipse $\partial E_{\rho}(\sqrt{s},0)$ in the framework $\mathscr{H}_{B}(\mathbb{C})$. More precisely, we have a Hilbert space isomorphism*

[TABLE]

whose inverse is given by

[TABLE]

and for $n=0,1,2,\dotsc$,

[TABLE]

Moreover, the reproducing kernel $K_{s}(z,\zeta)$ of $\mathscr{X}_{s}(\mathbb{C})$ can be also obtained by the Hilbert space isomorphism and the reproducing formula for $\mathscr{H}_{B}(\mathbb{C})$.

Recall the definition of $\partial E_{\rho}(\sqrt{s},0)$, that is,

[TABLE]

We here remark that $\{\partial E_{\rho}(\sqrt{s},0)\ |\ 0<s<1,\rho>0\}$ is the set of all ellipses whose major and minor axes are contained in the real and imaginary axes respectively.

Proof of Theorem 3.

First we shall show that (1) is a Hilbert space isomorphism of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$ and its inverse is given by (2). Let $\varphi(z)$ be a Lebesgue measurable function on $\mathbb{C}$. Set $P(z)=\varphi(z)e^{z^{2}/2}$, which corresponds to the holomorphic Hermite polynomials $H_{n}(z)$. By using an identity of the form

[TABLE]

we deduce that

[TABLE]

If we change the variable by

[TABLE]

then we have

[TABLE]

If $\varphi\in\mathscr{X}_{s}(\mathbb{C})$, then

[TABLE]

is holomorphic in $\mathbb{C}$ and belongs to $\mathscr{H}_{B}(\mathbb{C})$. Hence (5) shows that (1) is an injective and isometric mapping of $\mathscr{X}_{s}(\mathbb{C})$ to $\mathscr{H}_{B}(\mathbb{C})$. In the same way one can show that (2) is an inverse of (1). Thus (1) is a Hilbert space isomorphism of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$, and its inverse is given by (2).

Next we show that $\mathscr{X}_{s}(\mathbb{C})$ and $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ are essentially determined by the ellipse $\partial E_{\rho}(\sqrt{s},0)$ in the framework $\mathscr{H}_{B}(\mathbb{C})$. This follows from (4) and the correspondence (3). For this reason, it suffices to show the correspondence (3). If we choose $(\alpha,\beta)=(\sqrt{s},0)$, then we have for $n=0,1,2,\dotsc$,

[TABLE]

[TABLE]

By using this, we deduce that

[TABLE]

which is a desired equation (3).

Finally, we show that the reproducing kernel $K_{s}(z,\zeta)$ can be obtained by the reproducing formula $\varphi=B{\circ}B^{\ast}\varphi$ for $\varphi\in\mathscr{H}_{B}(\mathbb{C})$. Let $\varphi\in\mathscr{X}_{s}(\mathbb{C})$. Then

[TABLE]

Substitute this into the reproducing formula for $\mathscr{H}_{B}(\mathbb{C})$. Then we have

[TABLE]

Hence,

[TABLE]

By using the change of variable

[TABLE]

for $z$ and $\zeta$, we deduce that

[TABLE]

This completes the proof. ∎

3. $\mathscr{X}_{s}(\mathbb{C})$ and the Bargmann-type transforms

In this section we study the function space $\mathscr{X}_{s}(\mathbb{C})$ and some related topics from a view point of Sjöstrand’s theory of microlocal analysis based on the Bargmann-type transforms. We can choose the phase function $\phi(z,x)$ so that

[TABLE]

Indeed, if the constants $a$, $b$ and $c$ satisfy

[TABLE]

then (6) holds. There are uncountably many choices of the triple $(a,b,c)$ satisfying (7). For example, the choice of

[TABLE]

satisfies the condition (6). Moreover, if the condition (7) is satisfied, then

[TABLE]

and

[TABLE]

Thus we have just proved the following theorem.

Theorem 4**.**

If we choose $(a,b,c)$ satisfying (7), then $\mathscr{X}_{s}(\mathbb{C})=\mathscr{H}_{\Phi}(\mathbb{C})$ and

[TABLE]

The reproducing kernel Hilbert space $\mathscr{X}_{s}(\mathbb{C})$ and the system of holomorphic Hermite functions $\{\psi^{s}_{n}\}_{n=0}^{\infty}$ have been applied to studying quantization on the complex plane, counting and combinatorics, and etc. In the study of quantization on $\mathbb{C}$, Twareque Ali, Górska, Horzela and Szafraniec constructed a Hilbert space isomorphism of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$ and its inverse concretely. More precisely, they constructed an integral transform of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$. They obtained its integral kernel by using the generating function of Hermite functions. Their idea works well since a system of monomials $\{z^{n}/\sqrt{\pi 2^{n+1}n!}\}_{n=0}^{\infty}$ is a complete orthonormal system of $\mathscr{H}_{B}(\mathbb{C})$. See their paper [5, Section III] for the detail.

Recall that $B:L^{2}(\mathbb{R})\rightarrow\mathscr{H}_{B}(\mathbb{C})$ and $T:L^{2}(\mathbb{R})\rightarrow\mathscr{H}_{\Phi}(\mathbb{C})$ are Hilbert space isomorphisms. By using this fact, one can construct uncountably many Hilbert space isomorphisms of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$. Then we have the following theorem.

Theorem 5**.**

If we choose $(a,b,c)$ satisfying (7), then $B{\circ}T^{\ast}$ is a Hilbert space isomorphism of $\mathscr{X}_{s}(\mathbb{C})$ onto $\mathscr{H}_{B}(\mathbb{C})$.

Finally we will see some examples of Theorem 5.

Theorem 6**.**

• •

If $a=i/s$, $b=-i\sqrt{1-s^{2}}$ and $c=is$, then $\mathscr{X}_{s}(\mathbb{C})=\mathscr{H}_{\Phi}(\mathbb{C})$, and for any $\psi\in\mathscr{X}_{s}(\mathbb{C})$

[TABLE]

• •

If $a=is$, $b=\sqrt{1-s^{2}}$ and $c=is$, then $\mathscr{X}_{s}(\mathbb{C})=\mathscr{H}_{\Phi}(\mathbb{C})$, and for any $\psi\in\mathscr{X}_{s}(\mathbb{C})$

[TABLE]

Proof.

We will check only the first example. The second one can be proved in the same way. We here omit the detail. Suppose that $a=i/s$, $b=i\sqrt{1-s^{2}}$ and $c=is$. Then

[TABLE]

We will reduce the explicit formula of $B{\circ}T^{\ast}\psi$. Applying $B$ on $T^{\ast}\psi$, we have

[TABLE]

Since

[TABLE]

we deduce that

[TABLE]

This completes the proof. ∎

Acknowledgements

The author is very grateful to Professor Franciszek Hugon Szafraniec for being interested in the manuscript [7] and for teaching the author recent topics on holomorphic Hermite functions kindly. This was a chance to write the present paper since the author knew nothing about them.