Rearranged Fourier Series and Generalizations to Non-Commutative Groups

Abstract

It is well-known that the Fourier series of continuous functions on the torus are not always uniformly convergent. However, P. L. Ulyanov proposed a problem: can we permute the Fourier series of each individual continuous function in such a way as to guarantee uniform convergence of the rearranged Fourier series? This problem remains open, but nonetheless a rather strong partial result was proved by S. G. Revesz which states that for every continuous function there exists a subsequence of rearranged partial Fourier sums converging to the function uniformly.We give several new equivalences to Ulyanov’s problem in terms of the convergence of the rearranged Fourier series in the strong and weak operator topologies on the space of bounded operators on L 2 ($\mathbb{T}$). This new approach gives rise to several new problems related to rearrangement of Fourier series. We also consider Ulyanov’s problem and Revesz’s theorem for reduced C*–algebras on discrete countable groups.