An Operator theoretic approach to the convergence of rearranged Fourier series

TL;DR

This paper explores the convergence of rearranged Fourier series using operator theory, providing new equivalences, frameworks, and characterizations that advance understanding of Ulyanov's conjecture on uniform convergence.

Contribution

It introduces a novel operator theoretic framework for analyzing Fourier series rearrangements, offering new equivalences and characterizations related to Ulyanov's conjecture.

Findings

New equivalences for Ulyanov's conjecture in operator topologies

Characterizations of unconditional convergence in SOT and WOT

Framework for analyzing convergence on subspaces of L2

Abstract

This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on $B(L_2(T))$. Additionally, a new framework for further investigation is introduced by considering convergence for subspaces of $L_2$, which leads to many methods for attempting to prove or disprove Ulyanov's conjecture. In this framework, we provide characterizations of unconditional convergence of the Fourier series in the SOT and WOT. These considerations also give rise to some interesting questions regarding weaker versions of the rearrangement problem. Along the way, we consider some interesting questions related to the classical theory of trigonometric polynomials. All of the results here admit natural extensions to arbitrary dimensions.