Convergence of $\sum c_k f(k x)$ and the Lip $\alpha$ class
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- by Christoph Aistleitner
- Proc. Amer. Math. Soc. 140 (2012), 3893-3903
- DOI: https://doi.org/10.1090/S0002-9939-2012-11237-7
- 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 \begin{equation} \sum _{k =1}^\infty c_k^2 < \infty . \end{equation} Gaposhkin generalized this result to series of the form \begin{equation} \sum _{k=1}^\infty c_k f(kx) \end{equation} 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 \begin{equation} \sum _{k=1}^\infty c_k^2 k^{1-2 \alpha } (\log k)^\beta < \infty \qquad \textrm {for some} \qquad \beta >1+2\alpha . \end{equation} 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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Bibliographic Information
- Christoph Aistleitner
- Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
- Email: aistleitner@math.tugraz.at
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3893-3903
- MSC (2010): Primary 42A61, 42A20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11237-7
- MathSciNet review: 2944730