Besicovitch and other generalizations of Bohr ’ s almost periodic functions

We discuss several classes of almost periodic functions which generalize the uniformly continuous almost periodic (a.p.) functions originally defined by Harald Bohr. We have two goals, which we accomplish in two ways from two different sources. The first is to develop the proper setting for a Riesz-Fischer type theorem for almost periodic functions, which we do by following [3], and which leads to the definition of the Besicovitch almost periodic functions. Then we switch in section 5 to showing that Besicovitch functions are naturally occurring, but by defining such functions on the set of nonnegative numbers. For this we follow [2]. This note is just a summary of results with few proofs. A good reference on the subtle differences between the plethora of definitions of a.p. functions is [1]. Warning: Table 2 at the end of section 6 of that paper is an excellent summary of what the authors show, but there is a lone arrow that points up and to the left that should point down to the right.