Smoothness and time frequency analysis in Sobolev-Besicovitch spaces of almost periodic functions

Abstract

Here, smoothness analysis of almost periodic functions is studied. Analogously to the case of L 2 (R), the smoothness of the class of Besicovitch almost periodic functions is measured in a classic form by controlling, in some sense, the increments f(x + h)− f(x) and in a dual form by the decay of its FourierBohr transform or by its approximation properties. The same problem is also treated considering the time-frequency representation given by the Gabor transform. Some results are given as equivalence of norms between appropriate function spaces.