Estimates for mean-square norms of functions with lacunary Fourier series

Abstract

We consider the properties of functions f from the space L²(T) on the period T = [−π, π) with lacunary Fourier series such that the size of each gap is not less than a given positive integer q − 1. We find two-sided estimates of the L² norms of such functions on T in terms of similar norms (more exactly, seminorms) on intervals I of length |I| = 2h < 2π. The estimates are obtained in terms of best one-sided integral approximations to the characteristic function of the interval (−h, h) by trigonometric polynomials of degree at most q−1. The issue considered in this paper appeared first in N. Wiener’s studies (1934). Important results in this area were obtained by A.E. Ingham (1936) and by A. Selberg in the 1970s.