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Brezis's Open Problem 5.6 on Universal Fourier Summation

Analysis · posed by Haim Brezis · partial

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Statement

Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑nσn,εn∣f^(n)∣2→deg⁡f\sum_n \sigma_{n,\varepsilon} n |\hat f(n)|^2 \to \deg f hold for Holder maps below the threshold? No. For every 0<α<1/30 < \alpha < 1/3 there is an f∈C0,α(S1;S1)f \in C^{0,\alpha}(S^1;S^1) for which the sum fails to converge to deg⁡f\deg f, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all p>3p > 3. The endpoint C0,1/3C^{0,1/3} is left unresolved.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Michal Cieszynski, using ChatGPT

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    ai co developed
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    The disclosure states that ChatGPT generated preliminary drafts of the proofs in the manuscript. The author then verified each argument in detail, revised the proofs where necessary, checked the cited sources, determined the final formulation of all results, and takes sole responsibility for the content.

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