On the derivative of a Bohr's almost periodic function (Q2752309)

If a function \(f: \mathbb{R}\to\mathbb{R}\) is almost periodic in Bohr's sense, while its derivative is continuous according to the metrics \(S^p\) (Stepanov has used this Weyl's metric in defining the \(S^p\)-almost periodic functions), then \(f'\) is an \(S^p\)-almost periodic function. Variations on this theme are given in other theorems, dealing with \(C^{(n)}\) metric (continuous functions on \(\mathbb{R}\), bounded together with their \(n\) derivatives and sup norm), almost periodic functions in variation and another type of almost periodicity defined by the author.