Paley-Wiener Theorems with respect to the spectral parameter

TL;DR

This paper explores Paley-Wiener theorems in the context of harmonic analysis on manifolds and homogeneous spaces, focusing on the spectral parameter and the image of smooth functions under Fourier transform.

Contribution

It provides an overview of Fourier analysis on G/K spaces and investigates the description of function spaces via spectral parameters for various geometric settings.

Findings

Characterization of the Fourier transform image on G/K spaces

Analysis of spectral parameterization for Euclidean motion group

Extension to semisimple symmetric spaces and limits

Abstract

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of the set of irreducible representations in the support of the Plancherel measure for L^2(M). We then discuss the Euclidean motion group, semisimple symmetric spaces, and some limits of those spaces.