Segal-Bargmann Transform and Paley-Wiener Theorems on Motion Groups

TL;DR

This paper explores the Segal-Bargmann transform on motion groups, providing characterizations of Poisson integrals and establishing a Paley-Wiener theorem using complexified representations, advancing harmonic analysis on these groups.

Contribution

It introduces new characterizations of Poisson integrals and proves a Paley-Wiener theorem for motion groups, extending classical analysis results to this setting.

Findings

Characterization of Poisson integrals on motion groups

Establishment of a Paley-Wiener theorem using complexified representations

Extension of harmonic analysis tools to Rn n K groups

Abstract

We study the Segal-Bargmann transform on a motion group Rn n K; where K is a compact subgroup of SO(n): A characterization of the Poisson integrals associated to the Laplacian on Rn n K is given. We also establish a Paley-Wiener type theorem using the complexified representations.