On convergence of Fourier series of Besicovitch almost periodic functions

The paper deals with convergence of the Fourier series of q-Besicovitch almost periodic functions of the form $$ f(t)\sim \mathop{\sum}\limits_{m=1}^{\infty }{a_m}{{\mathrm{e}}^{{-\mathrm{i}{\uplambda_m}t}}}, $$ where {λm} is a Dirichlet sequence, that is, a strictly increasing sequence of nonnegative numbers tending to infinity. In particular, we show that, for 1 < q < ∞, the Fourier series of f(t) converges in norm to the function f(t) itself with usual order, which is analogous to the convergence in norm of the Fourier series of a function on 0, 2π]. A version of the Carleson–Hunt theorem is also investigated.