A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces

TL;DR

This paper develops a discrete Helgason-Fourier transform for functions on noncompact symmetric spaces, enabling exact and approximate computations for Sobolev and Besov functions using sampled data.

Contribution

It introduces a discrete Helgason-Fourier transform formula for Paley-Wiener functions and develops an approximation theory for Besov space functions on symmetric spaces.

Findings

Exact formula for Helgason-Fourier transform from sampled data

Discrete approximation methods for Besov space functions

Theoretical foundation for numerical analysis on symmetric spaces

Abstract

Let $f$ be a Paley-Wiener function in the space $L_{2}(X)$, where $X$ is a symmetric space of noncompact type. It is shown that by using the values of $f$ on a sufficiently dense and separated set of points of $X$ one can give an exact formula for the Helgason-Fourier transform of $f$. In order to find a discrete approximation to the Helgason-Fourier transform of a function from a Besov space on $X$ we develop an approximation theory by Paley-Wiener functions in $L_{2}(X)$.