ProbXiv
sign in
unchecked

Brezis's Open Problem 5.6 on Universal Fourier Summation

Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.

brezis-open-problem-5-6-fourier-summationAnalysisposed by Haim Brezisrecorded: partial

1 attempt · no person has looked

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.

Context

negative below the 1/3 threshold; the endpoint case is still open

A numbered entry on Brezis's published list of favourite open problems, a recognized problem list in analysis, with the degree-and-Fourier-moduli circle of questions attached to it.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionChatGPT with Michal Cieszynski ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT
    people
    Michal Cieszynski

    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.

    Reviews

    0 human reviews · 0 machine checks

    No person has reviewed this attempt. It has not been checked at all.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.