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On Wiener's Lemma on locally compact abelian groups

Philippe Jaming, Karim Kellay, Rolando Perez

TL;DR

This paper extends Wiener's lemma to measures on locally compact abelian groups by marrying Fourier analysis with Følner sequences, producing a unified framework that covers both discrete and continuous settings. It proves a general lemma expressing the mass of the point part $\mu(\{\mathbf{1}_G\})$ as limits of Fourier-domain or spatial averages, and it establishes Wiener-type identities along Følner sequences in $\widehat{G}$. The authors then apply these ideas to weighted means on $\mathbb{R}^d$ and $\mathbb{T}^d$, showing how the discrete part of a finite measure can be reconstructed from Bochner–Riesz-type kernels, including Gaussian and power-type weights. The results yield concrete formulas linking $\mu(\{x\})$ to smoothed or truncated Fourier transforms, with implications for both harmonic analysis on LCA groups and summation methods via Bochner–Riesz means on Euclidean and toral settings.

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{ø}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.

On Wiener's Lemma on locally compact abelian groups

TL;DR

This paper extends Wiener's lemma to measures on locally compact abelian groups by marrying Fourier analysis with Følner sequences, producing a unified framework that covers both discrete and continuous settings. It proves a general lemma expressing the mass of the point part as limits of Fourier-domain or spatial averages, and it establishes Wiener-type identities along Følner sequences in . The authors then apply these ideas to weighted means on and , showing how the discrete part of a finite measure can be reconstructed from Bochner–Riesz-type kernels, including Gaussian and power-type weights. The results yield concrete formulas linking to smoothed or truncated Fourier transforms, with implications for both harmonic analysis on LCA groups and summation methods via Bochner–Riesz means on Euclidean and toral settings.

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{ø}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.
Paper Structure (7 sections, 5 theorems, 42 equations)

This paper contains 7 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Results
  4. A General Lemma
  5. Fø lner sequences
  6. Weighted means on $\mathbb{R}^d$
  7. Weighted means on $\mathbb{T}^d$

Key Result

Lemma 3.1

Let $\mu$ be a complex-valued Borel measure on a locally compact abelian group $G$. For $R>0$, let $(\varphi_R)_{R>0}$ be a family of continuous functions on $G$ such that Then Furthermore, if $\varphi_R$ is such that there exists a $\psi_R\in L^1(\widehat{G})$ with $\varphi_R=\mathcal{F}_{\widehat{G}}[\psi_R]$, then

Theorems & Definitions (13)

  • Lemma 3.1
  • proof
  • Theorem 3.2: Wiener's lemma for Fø lner sequences
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • proof
  • Example 3.7
  • ...and 3 more