Segal-Bargmann-Fock modules of monogenic functions

TL;DR

This paper extends the Segal-Bargmann transform to Clifford algebra-valued monogenic functions, establishing connections with the Fourier-Borel transform and reproducing kernels in the context of monogenic modules.

Contribution

It introduces a Clifford algebra-valued Segal-Bargmann transform and links it to monogenic functions and their kernels, expanding classical analysis tools to Clifford analysis.

Findings

Segal-Bargmann kernel matches Fourier-Borel kernel for monogenic functionals

Kernel serves as the reproducing kernel for monogenic Bargmann modules

Extension of classical transform to Clifford algebra-valued functions

Abstract

In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.