重排傅里叶级数收敛性的算子论方法
An Operator theoretic approach to the convergence of rearranged Fourier series
摘要：
本文研究了P.L.提出的傅里叶级数的重排问题。Ulyanov，谁阐明，每一个连续函数的环面承认重新安排其傅里叶系数，使重新安排的部分和的傅里叶级数收敛一致的功能。这里的主要定理给出了这个猜想的几个新的等价条件，即重排的傅里叶级数在$B(L_2(T))$上的强（等价于在这种情况下，弱）算子拓扑中的收敛性。此外，通过考虑$L_2$的子空间的收敛性，引入了一个新的研究框架，从而引出了许多试图证明或证伪Ulyanov猜想的方法.在这个框架下，我们提供了傅里叶级数在SOT和WOT中的无条件收敛的特征。这些考虑也引起了一些有趣的问题，关于较弱版本的重排问题。一路上，我们考虑一些有趣的问题有关的经典理论的三角多项式。这里的所有结果都允许对任意维数的自然扩张。
英文摘要：
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.
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