Brezis's Open Problem 5.6 on Universal Fourier Summation
Prior state unknown→disproved
negative below the 1/3 threshold; the endpoint case is still open
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analysis / Harmonic analysis
Brezis's Open Problem 5.6 on Universal Fourier Summation
Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\sum_n \sigma_{n,\varepsilon} n |\hat f(n)|^2 \to \deg f$ hold for Holder maps below the threshold? No. For every $0 < \alpha < 1/3$ there is an $f \in C^{0,\alpha}(S^1;S^1)$ for which the sum fails to converge to $\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.
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Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\sum_n \sigma_{n,\varepsilon} n |\hat f(n)|^2 \to \deg f$ hold for Holder maps below the threshold? No. For every $0 < \alpha < 1/3$ there is an $f \in C^{0,\alpha}(S^1;S^1)$ for which the sum fails to converge to $\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.
negative below the 1/3 threshold; the endpoint case is still open
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