Eigenfunction of the Laplacian as a Degenerate Case of a Function with its Fourier Transform Supported in an Annulus

Abstract

The aim of this paper is to reach a characterization of the eigenfunction of the Laplace–Beltrami operator as a degenerate case of the inverse Paley–Wiener theorem (for functions whose Fourier transform is supported on a compact annulus) on the rank-1 Riemannian symmetric spaces of noncompact type. The most distinguished prototypes of these spaces are the hyperbolic spaces.