Abstract
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.
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Acknowledgements
The authors thank Szilárd Révész, Rudy Rodsphon, Alex Powell, Alexander Olevskii and Artur Sahakian for valuable discussions related to this work. We also thank the anonymous referees for valuable suggestions.
Part of this work was done while the first author was an Assistant Professor at Vanderbilt University. The first author acknowledges partial support in the later stages from the National Science Foundation TRIPODS program, grant CCF-1740858.
Part of this work was done while the second author was an Assistant Professor at Vanderbilt University. The second author acknowledges support from the grants NSF DMS-1600802 and NSF DMS-1827376.
Part of the work of this paper was done while the third author was a Ph.D. student at Vanderbilt University; the project was motivated by a question asked to him by Vaughan Jones during a class. The third author acknowledges support during the final stages of the project by the Oak Ridge National Laboratory, which is operated by UT-B attelle, LLC, for the U.S. Department of Energy under Contract DE-AC05-00OR22725.
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Hamm, K., Hayes, B. & Petrosyan, A. An Operator theoretic approach to the convergence of rearranged Fourier series. JAMA 143, 503–534 (2021). https://doi.org/10.1007/s11854-021-0161-8
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DOI: https://doi.org/10.1007/s11854-021-0161-8