On the Convergence of Symbol C Kf(n Kx)

Let f be a periodic measurable function and x (nk) an increasing sequence of positive integers. The authors study conditions under which the series k=1 Ckf(nkx)_ converges in mean and for almost every x. There is a wide classical literature on this problem going back to the 30's, but the results for general f are much less complete than in the trigonometric case f(x) = sin x. As it turns out, the convergence properties of k=1 ckf(nkx) in the general case are determined by a delicate interplay between the coefficient sequence (ck), the analytic properties of f and the growth speed and number-theoretic properties of (nk). In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the nk are random and investigate the discrepancy the sequence {nkx} mod 1.