This article studies the rearrangement problem for Fourier series introduced by P. L. Ulyanov, who posed the question if 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 problem in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on \({\cal B}({L_2}(\mathbb{T}))\)). Additionally, a new framework for further investigation is introduced by considering convergence for subspaces of L₂, which leads to many methods for attempting to prove or disprove Ulyanov’s problem. 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.

本文研究了 PL Ulyanov 引入的傅里叶级数重排问题，他提出了一个问题，即环面上的每个连续函数是否都允许重排其傅里叶系数，使得傅里叶级数的重排部分和均匀收敛到该函数。这里的主要定理根据重新排列的傅立叶级数在\({\cal B}({L_2}(\mathbb{ T}))\) )。此外，通过考虑L _{2 的}子空间的收敛性，引入了用于进一步研究的新框架，这导致了许多试图证明或反驳 Ulyanov 问题的方法。在这个框架中，我们提供了 SOT 和 WOT 中傅立叶级数无条件收敛的特征。这些考虑也引起了一些关于重排问题的较弱版本的有趣问题。在此过程中，我们考虑了一些与三角多项式经典理论相关的有趣问题。这里的所有结果都允许自然扩展到任意维度。