On composition of Segal-Bargmann transforms

TL;DR

This paper explores integral transforms within various Bargmann-Fock spaces, analyzing their properties, relationships, and inverses, and connecting complex and quaternionic function spaces through these transforms.

Contribution

It introduces new integral transforms linking different Bargmann-Fock spaces and studies their properties, inverses, and connections to quaternionic hyperholomorphic spaces.

Findings

Identified the image of the integral transforms in Bargmann-Fock spaces.

Determined left-inverses for the transforms on these spaces.

Established connections between complex and quaternionic hyperholomorphic function spaces.

Abstract

We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on $\mathbb{C}$ and $\mathbb{C}^2$ and the slice hyperholomorphic Bargmann-Fock space on $\mathbb{H}$. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the two-dimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.