On Wiener's Lemma on locally compact abelian groups

TL;DR

This paper generalizes Wiener's lemma for measures on locally compact abelian groups using Fourier analysis and F{ exto}lner sequences, unifying discrete and continuous cases, and extends it to Bochner-Riesz means on Euclidean and torus spaces.

Contribution

It introduces a unified framework for Wiener's lemma on LCA groups and extends the results to Bochner-Riesz means on R^d and T^d.

Findings

Unified Wiener's lemma for LCA groups

Extension to Bochner-Riesz means on R^d and T^d

Bridging discrete and continuous cases in harmonic analysis

Abstract

We establish a general form of Wiener's lemma for measures on locally compact abelian (LCA) groups by using Fourier analysis and the theory of F{{\o}}lner sequences. Our approach provides a unified framework that that encompasses both the discrete and continuous cases. We also show a version of Wiener's lemma for Bochner-Riesz means on both R^d and T^d . Mathematics Subject Classification (2010). 43A25.