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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convergence of $ {\sum}c_k f(k x)$ and the Lip $ \alpha$ class


Author: Christoph Aistleitner
Journal: Proc. Amer. Math. Soc. 140 (2012), 3893-3903
MSC (2010): Primary 42A61, 42A20
Published electronically: March 21, 2012
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Abstract: By Carleson's theorem a trigonometric series $ \sum _{k=1}^\infty c_k \cos 2 \pi k x$ or $ \sum _{k=1}^\infty c_k \sin 2 \pi k x$ is a.e. convergent if

$\displaystyle \sum _{k =1}^\infty c_k^2 < \infty .$ (1)

Gaposhkin generalized this result to series of the form

$\displaystyle \sum _{k=1}^\infty c_k f(kx)$ (2)

for functions $ f$ satisfying $ f(x+1)=f(x),~\int _0^1 f(x)=0$ and belonging to the Lip $ \alpha $ class for some $ \alpha >1/2$. In the case $ \alpha \leq 1/2$ condition (1) is in general no longer sufficient to guarantee the a.e. convergence of (2).

For $ 0 < \alpha < 1/2$ Gaposhkin showed that (2) is a.e. convergent if

$\displaystyle \sum _{k=1}^\infty c_k^2 k^{1-2 \alpha } (\log k)^\beta < \infty \qquad \textrm {for some} \qquad \beta >1+2\alpha .$ (3)

In this paper we show that condition (3) can be significantly weakened for $ \alpha \in [1/4,1/2)$. In fact, we show that in this case the factor $ k^{1-2\alpha }(\log k)^\beta $ can be replaced by a factor which is asymptotically smaller than any positive power of $ k$.

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Additional Information

Christoph Aistleitner
Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Email: aistleitner@math.tugraz.at

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11237-7
PII: S 0002-9939(2012)11237-7
Keywords: Almost everywhere convergence, Lipschitz classes
Received by editor(s): June 17, 2010
Received by editor(s) in revised form: March 7, 2011, and May 9, 2011
Published electronically: March 21, 2012
Additional Notes: This research was supported by the Austrian Research Foundation (FWF), Project S9603-N23. This paper was written while the author was a participant of the Oberwolfach Leibniz Fellowship Programme (OWLF) of the Mathematical Research Institute of Oberwolfach, Germany.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.