Segal-Bargmann transform and Paley-Wiener theorems on $M(2).$

TL;DR

This paper investigates the Segal-Bargmann transform on the group M(2), characterizing its range as a weighted Bergman space, and establishes Paley-Wiener theorems related to Fourier transforms on this group.

Contribution

It provides a new characterization of the Segal-Bargmann transform's range on M(2) and proves a Paley-Wiener theorem for the inverse Fourier transform on this group.

Findings

Range characterized as weighted Bergman space

Holomorphic extension via Gutzmer formula

Paley-Wiener theorem established for inverse Fourier transform

Abstract

We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform