We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class
of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients
We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with
general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.
Let (nk)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying nk+1/nk > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝01f(x) dx = 0. Then the probabilistic behavior of the system (f(nkx))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following
law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary
for almost all x ∈ (0, 1), where ‖f‖2 = (∝01f(x)2dx)1/2 is the standard deviation of the random variables f(nkx). If (nk)k≧1 has certain number-theoretic properties (e.g. nk+1/nk → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖2. For general lacunary (nk)k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (nk)k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖2 and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can
be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (nk)k≧1 such that (1) holds with √‖f‖22 + g(x) instead of ‖f‖2 on the right-hand side.