Some remarks on Bohr's almost periodic functions and Stepanov's almost periodic functions (Q1370576)

There are considered V-a.p. functions and L-a.p. functions \(x\) on the real line (i.e., almost periodic in the sense of variation and in the sense of Lipschitz condition, respectively). It is proved that if \(x\) is S-a.p. and the indefinite integral of \(x\) is bounded and uniformly continuous, then this integral is V-a.p. Moreover, if \(x\) is u.a.p., then its integral means \(S_x(u,h)= {1\over 2h} \int^{u+ h}_{u- h}x(s)ds\) are L-a.p. for every \(u\in\mathbb{R}\) and \(S_x(u, h)\to x(u)\) as \(h\to 0\) uniformly for \(u\in\mathbb{R}\).