NDLI: On the class of limits of lacunary trigonometric series

Content Provider	Springer Nature Link
Author	Aistleitner, C.
Copyright Year	2010
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 ∝01 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 ‖f‖2 = (∝01 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 ‖f‖2. 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 ‖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 (n k)k≧1 such that (1) holds with √‖f‖22 + g(x) instead of ‖f‖2 on the right-hand side.
Ending Page	23
Page Count	23
Starting Page	1
File Format	PDF
ISSN	02365294
e-ISSN	15882632
Journal	Acta Mathematica Hungarica
Issue Number	1-2
Volume Number	129
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-04-15
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	law of the iterated logarithm Irregularities of distribution, discrepancy lacunary series Lacunary series of trigonometric and other functions; Riesz products Mathematics Strong theorems
Content Type	Text
Resource Type	Article
Subject	Mathematics

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