Function with its Fourier transform supported on annulus and eigenfunction of Laplacian

TL;DR

This paper investigates the characterization of Laplacian eigenfunctions as a special case of the inverse Paley-Wiener theorem on noncompact Riemannian symmetric spaces, extending classical Fourier analysis results.

Contribution

It extends the inverse Paley-Wiener theorem to characterize Laplacian eigenfunctions on noncompact symmetric spaces, including hyperbolic and Damek-Ricci spaces.

Findings

Characterization of eigenfunctions via Fourier support on annuli

Extension of Paley-Wiener theorem to noncompact symmetric spaces

Applicable to Euclidean, hyperbolic, and Damek-Ricci spaces

Abstract

We explore the possibilities of reaching the characterization of eigenfunction of Laplacian as a degenerate case of the inverse Paley-Wiener theorem (characterizing functions whose Fourier transform is supported on a compact annulus) for the Riemannian symmetric spaces of noncompact type. Most distinguished prototypes of these spaces are the hyperbolic spaces. The statement and the proof of the main result work mutatis-mutandis for a number of spaces including Euclidean spaces and Damek-Ricci spaces.