The Dirichlet kernel on the real hyperbolic space for radial functions

TL;DR

This paper investigates the convergence properties of spherical partial integrals of Fourier transforms on real hyperbolic space, providing new asymptotic results and a unified analytical framework.

Contribution

It introduces a unified method to analyze convergence of Fourier transforms on hyperbolic space, yielding new asymptotic expansions and convergence criteria.

Findings

Asymptotic expansions for spherical partial integrals

Necessary and sufficient conditions for pointwise convergence

A unified framework connecting new and existing results

Abstract

Asymptotic expansions as well as necessary and sufficient conditions are provided for the pointwise convergence of the spherical partial integrals of the associated Fourier transforms on the real hyperbolic space. The proposed method permits one to produce new results and opens a connection with already known ones utilizing a unified framework.