The set of continuous functions with everywhere convergent fourier series

This paper deals with the descriptive set theoretic properties of the class 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 EC is studied that assigns to each f 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 EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with wi distinct levels. © 1987 American Mathematical Society.