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Acta Mathematica Hungarica
Volume/Issue:
Volume 114: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-006-0513-y
Author:
S. Tikhonov

Abstract  

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 are presented.

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Acta Mathematica Hungarica
Volume/Issue:
Volume 134: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-011-0092-4
Authors:
R. J. Le
and
S. P. Zhou

References [1] Kórus , P. 2010 Remarks on the uniform and L 1 -convergence of trigonometric series Acta Math. Hungar

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Acta Mathematica Hungarica
Volume/Issue:
Volume 120: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-008-7074-1
Author:
S. Tikhonov

Abstract  

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.

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Acta Mathematica Hungarica
Volume/Issue:
Volume 129: Issue 4
DOI:
https://doi.org/10.1007/s10474-010-0034-6
Author:
Katusi Fukuyama

References [1] Berkes , I. 1978 On the central limit theorem for lacunary trigonometric series Anal. Math

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Periodica Mathematica Hungarica
Volume/Issue:
Volume 21: Issue 4
DOI:
https://doi.org/10.1007/bf02352692
Author:
I. Berkes

Without Abstract

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Analysis Mathematica
Volume/Issue:
Volume 16: Issue 3
DOI:
https://doi.org/10.1007/bf01906082
Author:
М. И. ДьяЧенко
В статье рассматрива ютсяn-кратные тригонометрические ряды вида(1)
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop \Sigma \limits_{k = 1}^\infty a_k \exp (ikx),$$ \end{document}
гдеa k ≧a m , еслиk j ≦m j при 1≦j≦n иa k→0 при maxk j →∞. Для таких рядов доказ ыва1≦j≦n ется несколько теорем, обо бщающих ранее получе нные автором утверждения дляи=2. Сформулируем две из н их.
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Acta Mathematica Hungarica
Volume/Issue:
Volume 57: Issue 1-2
DOI:
https://doi.org/10.1007/bf01903816
Author:
I. Berkes
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Acta Mathematica Hungarica
Volume/Issue:
Volume 47: Issue 1-2
DOI:
https://doi.org/10.1007/bf01949133
Author:
N. Tanović-Miller
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Acta Mathematica Hungarica
Volume/Issue:
Volume 129: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-010-9218-3
Author:
C. Aistleitner

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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(n_k )_{k \geqq 1}$$ \end{document}
:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2$$ \end{document}
((1)) for almost all x ∈ (0, 1), where ‖f‖2 = (∝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 ‖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‖2 2 + g(x) instead of ‖f‖2 on the right-hand side.
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Periodica Mathematica Hungarica
Volume/Issue:
Volume 46: Issue 2
DOI:
https://doi.org/10.1023/a:1025983909311
Authors:
L. Bernal-González
,
M. Calderón-Moreno
, and
W. Luh
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