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

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 $k$.