Paley-Wiener-Schwartz nearly Parseval frames and Besov spaces on noncompact symmetric spaces

TL;DR

This paper constructs nearly Parseval frames on noncompact symmetric spaces that belong to both Paley-Wiener and Schwartz spaces, enabling characterization of Besov spaces through a novel sampling method.

Contribution

It introduces Paley-Wiener-Schwartz frames on noncompact symmetric spaces and develops average Shannon-type sampling for these spaces.

Findings

Frames are nearly Parseval and belong to Paley-Wiener and Schwartz spaces.

Frames characterize Besov spaces on the symmetric space.

Develops a new sampling method for analysis on noncompact symmetric spaces.

Abstract

Let $X$ be a symmetric space of the noncompact type. The goal of the paper is to construct in the space $L_{2}(X)$ nearly Parseval frames consisting of functions which simultaneously belong to Paley-Wiener spaces and to Schwartz space on $X$. We call them Paley-Wiener-Schwartz frames in $L_{2}(X)$. These frames are used to characterize a family of Besov spaces on $X$. As a part of our construction we develop on $X$ the so-called average Shannon-type sampling.