Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the convergence of $\sum c_nf(nx)$ 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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42A55, 42A61

Retrieve articles in all journals with MSC (1991): 42A55, 42A61

Additional Information

István Berkes
Affiliation: Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary
MR Author ID: 35400

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