Fourier Representations in Bergman Spaces

TL;DR

This paper develops Fourier, Fourier series, and Mellin integral representations for functions in Bergman spaces of symmetric domains, enabling new ways to analyze Bergman kernels and their symmetries.

Contribution

It introduces Paley-Wiener type representations for Bergman spaces on generalized domains with symmetries, extending classical analysis tools.

Findings

Derived Fourier series, integral, and Mellin integral representations.

Provided new representations of Bergman kernels for these domains.

Extended classical harmonic analysis to more general symmetric domains.

Abstract

We consider a class of domains, generalizing the upper half-plane, and admitting rotational, translational and scaling symmetries, analogous to the half-plane. We prove Paley-Wiener type representations of functions in Bergman spaces of such domains with respect to each of these three groups of symmetries. The Fourier series, Fourier integral and Mellin integral representations so obtained may be used to give representations of the Bergman kernels of these domains.