We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
On the class of limits of lacunary trigonometric series.
- Authors
Aistleitner, C.
- Abstract
Let ( n) be a lacunary sequence of positive integers, i.e. a sequence satisfying n/ n > q > 1, k ≧ 1, and let f be a 'nice' 1-periodic function with ∝ f( x) dx = 0. Then the probabilistic behavior of the system ( f( n x)) 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 2 πx and lacunary $$ (n_k )_{k \geqq 1} $$: for almost all x ∈ (0, 1), where | f| = (∝ f( x) dx) is the standard deviation of the random variables f( n x). If ( n) has certain number-theoretic properties (e.g. n/ n → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always | f|. For general lacunary ( n) this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence ( n), such that the lim sup in the LIL (1) is not equal to | f| 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) such that (1) holds with √| f| + g( x) instead of | f| on the right-hand side.
- Subjects
FOURIER series; FOURIER analysis; MATHEMATICAL statistics; CLUSTER analysis (Statistics); STANDARD deviations
- Publication
Acta Mathematica Hungarica, 2010, Vol 129, Issue 1/2, p1
- ISSN
0236-5294
- Publication type
Article
- DOI
10.1007/s10474-010-9218-3