Pointwise universal trigonometric series

TL;DR

This paper investigates conditions under which trigonometric series can approximate any continuous function pointwise on the unit circle, identifying growth constraints on coefficients that enable or prevent universality.

Contribution

It establishes new growth criteria for coefficients that determine the existence of pointwise universal trigonometric series.

Findings

Series with coefficients growing slower than $e^{|n|/( ext{log}|n|)^{1+ ext{epsilon}}}$ cannot be universal.

Existence of universal series with coefficients growing as $e^{|n|/\text{log}|n|}$.

Provides a characterization of coefficient growth rates for universality.

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 $|a_n|=O(\e^{\,|n|\ln^{-1-\epsilon}|n|})$ as $|n|\to\infty$ for some $\epsilon>0$, 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 $|n|\to\infty$.