The (reflected) Eberlein convolution of measures

Abstract

In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected 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 (reflected) 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 (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution.