A theorem of Levinson for Riemannian symmetric spaces of noncompact type

TL;DR

This paper extends Levinson's classical theorem from the real line to Riemannian symmetric spaces of noncompact type, linking the vanishing of functions on open sets to decay properties of their Fourier transforms.

Contribution

It provides a new analogue of Levinson's theorem for a broader class of geometric spaces, specifically Riemannian symmetric spaces of noncompact type.

Findings

Established a Levinson-type theorem for symmetric spaces

Connected function vanishing properties to Fourier transform decay

Extended classical harmonic analysis results to non-Euclidean spaces

Abstract

A classical result of N. Levinson characterizes the existence of a nonzero integrable function vanishing on a nonempty open subset of the real line in terms of the pointwise decay of its Fourier transform. We prove an analogue of this result for Riemannian symmetric spaces of noncompact type.