The (twisted) Eberlein convolution of measures

TL;DR

This paper introduces a twisted version of the Eberlein convolution for measures, exploring its properties, regularity, and conditions under which it exhibits almost periodicity, with implications for harmonic analysis.

Contribution

It defines and analyzes a novel twisted Eberlein convolution, establishing its existence, regularity, and almost periodicity properties for measures and functions.

Findings

Twisted Eberlein convolution exists along subsequences for translation bounded measures.

It is weakly almost periodic and Fourier transformable.

Strong and norm almost periodicity are preserved under certain conditions.

Abstract

In this paper, we study the properties of the Eberlein convolution of measures and introduce a twisted version of it. For functions we show that the twisted Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (twisted) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (twisted) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (twisted) Eberlein convolution.