Asymptotic results on modified Bergman-Dirichlet spaces and examples of Segal-Bargmann transforms

TL;DR

This paper investigates the asymptotic behavior of modified Bergman-Dirichlet spaces as parameters vary, deriving new spaces and kernels, and provides examples of Segal-Bargmann transforms related to these spaces.

Contribution

It introduces modified Bergman-Dirichlet spaces, analyzes their asymptotics, and constructs examples of Segal-Bargmann transforms for these new spaces.

Findings

Asymptotic behavior of spaces as parameter α approaches infinity and -1

Derivation of modified Bargmann-Dirichlet and Hardy-Dirichlet spaces

Explicit examples of Segal-Bargmann transforms for these spaces

Abstract

In this paper, we start by introducing the modified Bergman-Dirichlet space $\mathcal D_m^2(\mathbb D_R,\mu^R_{\alpha,\beta})$ and then we study its asymptotic behavior when the parameter $\alpha$ goes to infinity and to $(-1)$ to obtain respectively the modified Bargmann-Dirichlet and the modified Hardy-Dirichlet spaces with their reproducing kernels. Finally, we give some examples of Segal-Bargmann transforms of those spaces.