On Wiener's lemma for locally compact abelian groups

2020/05/11 by Schulte, Eike
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2005.05212

Abstract

Inspired by an extension of Wiener's lemma on the relation of measures μ on the unit circle and their Fourier coefficients \widehatμ(k_n) along subsequences (k_n) of the natural numbers by Cuny, Eisner and Farkas [CEF19, arXiv:1701.00101], we study the validity of the lemma when the Fourier coefficients are weighted by a sequence of probability measures. By using convergence with respect to a filter derived from these measure sequences, we obtain similar results, now also allowing the consideration of locally compact abelian groups other than \mathbbT and ℝ. As an application, we present an extension of a result of Goldstein [Gol96] on the action of semigroups on Hilbert spaces.

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