2018/09/21 by Keaton Hamm, Ben J. Hayes, Hamm, Keaton +3
Mathematics · #42A20 #42A32 #46E15 #46E20 #46J10 #47L05 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1809.08217
openalex publication_date 2018/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article studies the rearrangement problem for Fourier series introduced\nby P.L. Ulyanov, who conjectured that every continuous function on the torus\nadmits a rearrangement of its Fourier coefficients such that the rearranged\npartial sums of the Fourier series converge uniformly to the function. The main\ntheorem here gives several new equivalences to this conjecture in terms of the\nconvergence of the rearranged Fourier series in the strong (equivalently in\nthis case, weak) operator topologies on B(L2(T)). Additionally, a new\nframework for further investigation is introduced by considering convergence\nfor subspaces of L2, which leads to many methods for attempting to prove or\ndisprove Ulyanov's conjecture. In this framework, we provide characterizations\nof unconditional convergence of the Fourier series in the SOT and WOT. These\nconsiderations also give rise to some interesting questions regarding weaker\nversions of the rearrangement problem.\n Along the way, we consider some interesting questions related to the\nclassical theory of trigonometric polynomials. All of the results here admit\nnatural extensions to arbitrary dimensions.\n