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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the convergence of $\sum c_{n} f(nx)$
and the Lip 1/2 class


Author: István Berkes
Journal: Trans. Amer. Math. Soc. 349 (1997), 4143-4158
MSC (1991): Primary 42A55, 42A61
MathSciNet review: 1401764
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Abstract: We investigate the almost everywhere convergence of $\sum c_{n} f(nx)$, where $f$ is a measurable function satisfying

\begin{equation*}f(x+1) = f(x), \qquad \int _{0}^{1} f(x) \, dx =0.\end{equation*}

By a known criterion, if $f$ satisfies the above conditions and belongs to the Lip $\alpha $ class for some $\alpha > 1/2$, then $\sum c_{n} f(nx)$ is a.e. convergent provided $\sum c_{n}^{2} < +\infty $. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions $f$ and almost exponentially growing sequences $(n_{k})$ such that $\sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $\sum c_{k}^{2} < +\infty $. For functions $f$ with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.


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Additional Information

István Berkes
Affiliation: Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary
Email: berkes@math-inst.hu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01837-0
PII: S 0002-9947(97)01837-0
Keywords: Almost everywhere convergence, Lipschitz classes, lacunary series, law of the iterated logarithm
Received by editor(s): March 27, 1996
Additional Notes: Research supported by Hungarian National Foundation for Scientific Research, Grants T 16384 and T 19346
Article copyright: © Copyright 1997 American Mathematical Society