The set of continuous functions with everywhere convergent Fourier series

Abstract:

This paper deals with the descriptive set theoretic properties of the class $\operatorname {EC}$ of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in $C(T)$. A natural coanalytic rank function on $\operatorname {EC}$ is studied that assigns to each $f \in \operatorname {EC}$ a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of $f$. It is shown that there exist functions in $\operatorname {EC}$ (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on $\operatorname {EC}$ with ${\omega _1}$ distinct levels.