On Wiener's lemma for locally compact abelian groups

Abstract: Inspired by an extension of Wiener's lemma on the relation of measures $\mu$ on the unit circle and their Fourier coefficients $\widehat{\mu}(k_n)$ 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 $\mathbb{T}$ and $\mathbb{R}$. As an application, we present an extension of a result of Goldstein [Gol96] on the action of semigroups on Hilbert spaces.