Non-trivial 1d and 2d integral transforms of Segal-Bargmann type

TL;DR

This paper introduces novel one- and two-dimensional integral transforms of Segal-Bargmann type using complex Hermite polynomials, connecting them to classical transforms and specific functional Hilbert spaces within the coherent states framework.

Contribution

It develops new integral transforms based on complex Hermite polynomials and explores their properties and connections to known transforms like Fourier and Wigner.

Findings

Defined two new Segal-Bargmann type transforms.

Established links to Fourier and Wigner transforms.

Analyzed properties and kernels related to special functions.

Abstract

Generating functions for the univariate complex Hermite polynomials (UCHP) are employed to introduce some non-trivial one and two-dimensional integral transforms of Segal-Bargmann type in the framework of specific functional Hilbert spaces. The approach used is issued from the coherent states framework. Basic properties of these transforms are studied. Connection to some special known transforms like Fourier and Wigner transforms is established. The first transform is a two-dimensional Segal-Bargmann transform whose kernel is related the exponential generating function of the UCHP. The second one connects the so-called generalized Bargmann-Fock spaces (or also true-poly-Fock spaces in the terminology of Vasilevski) that are realized as the $L^2$-eigenspaces of a specific magnetic Schr\"odinger operator. Its kernel function is related to a Mehler formula of the UCHP.