An Operator theoretic approach to the convergence of rearranged Fourier series
An Operator theoretic approach to the convergence of rearranged Fourier series
复制标题
重排傅里叶级数收敛的算子理论方法
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Armenak Petrosyan
中科院分区：
文献类型：
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作者：
Keaton Hamm;Ben Hayes;Armenak Petrosyan
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 ℬ(L2(T))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal B}({L_2}(\mathbb{T}))$$\end{document}). 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.
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 ℬ(L2(T))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal B}({L_2}(\mathbb{T}))$$\end{document}). 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.