, Volume 129, Issue 1-2, pp 1-23
Date: 15 Apr 2010

On the class of limits of lacunary trigonometric series

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Abstract

Let (n k ) k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ 0 1 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x)) 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 \( (n_k )_{k \geqq 1} \) : (1) $$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$ for almost all x ∈ (0, 1), where ‖f2 = (∝ 0 1 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k ) k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f2. For general lacunary (n k ) k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k ) k≧1, such that the lim sup in the LIL (1) is not equal to ‖f2 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 (n k ) k≧1 such that (1) holds with √‖f 2 2 + g(x) instead of ‖f2 on the right-hand side.

Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. This paper was written during a stay at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences, which was made possible by an MOEL scholarship of the Österreichische Forschungsgemeinschaft (ÖFG).