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New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions

Daniel Alpay, Antonino De Martino, Kamal Diki

TL;DR

The work characterizes how the Schwartz space $\mathcal{S}$ and its dual $\mathcal{S}'$ are mapped by the Segal-Bargmann transform, revealing reproducing kernel Hilbert spaces whose kernels are governed by Touchard polynomials and hypergeometric functions. It introduces two families of weighted Fock-type spaces, $\mathcal{H}_p(\mathbb{C})$ and $\mathcal{F}_p(\mathbb{C})$, and develops corresponding generalized Bargmann transforms $\mathcal{B}_p$ and $\mathcal{SB}_p$, with explicit kernel representations and unitary properties. A detailed operator framework is established, including adjoints and commutator structures for backward shifts, annihilation, momentum, and integration operators, along with a geometric perspective via a Stieltjes moment problem for $\mathcal{H}_p$ and the absence of such a description for $\mathcal{F}_p$. The two spaces are linked through the Fock space by the diagonal-operator-driven maps $\Theta_p=(\mathcal{I}+M_z\partial)^{2p}$ and $\Lambda_p=(R_0 I)^{2p}$, unifying Touchard and hypergeometric kernels and showing how $\mathcal{H}_p$ and $\mathcal{F}_p$ arise as interconnected realizations of Bargmann-type transforms.

Abstract

In this paper we study the ranges of the Schwartz space $\mathcal S$ and its dual $\mathcal S^\prime$ (space of tempered distributions) under the Segal-Bargmann transform. The characterization of these two ranges lead to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed respectively in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators

New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions

TL;DR

The work characterizes how the Schwartz space and its dual are mapped by the Segal-Bargmann transform, revealing reproducing kernel Hilbert spaces whose kernels are governed by Touchard polynomials and hypergeometric functions. It introduces two families of weighted Fock-type spaces, and , and develops corresponding generalized Bargmann transforms and , with explicit kernel representations and unitary properties. A detailed operator framework is established, including adjoints and commutator structures for backward shifts, annihilation, momentum, and integration operators, along with a geometric perspective via a Stieltjes moment problem for and the absence of such a description for . The two spaces are linked through the Fock space by the diagonal-operator-driven maps and , unifying Touchard and hypergeometric kernels and showing how and arise as interconnected realizations of Bargmann-type transforms.

Abstract

In this paper we study the ranges of the Schwartz space and its dual (space of tempered distributions) under the Segal-Bargmann transform. The characterization of these two ranges lead to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed respectively in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators
Paper Structure (12 sections, 58 theorems, 283 equations)

This paper contains 12 sections, 58 theorems, 283 equations.

Table of Contents

  1. Introduction and preliminary results
  2. The space $\mathcal{H}_p$ and associated operators
  3. Construction and basic properties of the $\mathcal{H}_p$-space
  4. Some operators on $\mathcal{H}_p$-space
  5. The $\mathcal{H}_p$-Bargmann transform
  6. The Fock space connected to the $\mathcal{H}_p$ space
  7. The space $\mathcal{F}_p$ and associated operators
  8. Construction and basic properties of the $\mathcal{F}_p$-space
  9. Some operators on $\mathcal{F}_p$-space
  10. The $\mathcal{F}_p$-Bargmann transform
  11. The Fock space connected to the $\mathcal{F}_p$-space
  12. A relation between the $\mathcal{H}_p$ and $\mathcal{F}_p$ spaces

Key Result

Lemma 2.1

For a fixed $p > 0$ we have where

Theorems & Definitions (127)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • ...and 117 more