Luzin's problem on Fourier convergence and homeomorphisms
Gady Kozma, Alexander Olevskii
TL;DR
The paper proves that for any continuous function on the circle, there exists an absolutely continuous homeomorphism making its Fourier series converge uniformly, solving a problem posed by Luzin.
Contribution
It establishes the existence of a homeomorphism that ensures uniform Fourier convergence for all continuous functions, addressing Luzin's longstanding problem.
Findings
Existence of an absolutely continuous homeomorphism for uniform Fourier convergence.
Resolution of Luzin's problem on Fourier series convergence.
Advancement in understanding the interplay between homeomorphisms and Fourier analysis.
Abstract
We show that for every continuous function there exists an absolutely continuous homeomorphism of the circle such that the Fourier series of the composition converges uniformly. This resolves a problem set by N. N. Luzin.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Analytic and geometric function theory · Meromorphic and Entire Functions