Pointwise universal trigonometric series

Abstract

A series $S_a=\sum\limits_{n=-\infty}^\infty a_nz^n$ is called a {\it pointwise universal trigonometric series} if for any $f\in C(\T)$, there exists a strictly increasing sequence $\{n_k\}_{k\in\N}$ of positive integers such that $\sum\limits_{j=-n_k}^{n_k} a_jz^j$ converges to $f(z)$ pointwise on $\T$. We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if $|a_n|=O(\e^{\,|n|\ln^{-1-ε}|n|})$ as $|n|\to\infty$ for some $ε>0$, then the series $S_a$ can not be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series $S_a$ with $|a_n|=O(\e^{\,|n|\ln^{-1}|n|})$ as $|n|\to\infty$.