Everywhere divergence of one-sided ergodic Hilbert transform
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2477-2500.

For a given number α(0,1) and a 1-periodic function f, we study the convergence of the series n=1 f(x+nα) n, called one-sided Hilbert transform relative to the rotation xx+αmod1. Among others, we prove that for any non-polynomial function of class C 2 having Taylor–Fourier series (i.e. Fourier coefficients vanish on - ), there exists an irrational number α (actually a residual set of α) such that the series diverges for all x. We also prove that for any irrational number α, there exists a continuous function f such that the series diverges for all x. The convergence of general series n=1 a n f(x+nα) is also discussed in different cases involving the diophantine property of the number α and the regularity of the function f.

Etant donné un nombre α(0,1) et une fonction 1-périodique f, nous étudions la convergence de la série n=1 f(x+nα) n, appelée la transformée de Hilbert latérale relative à la rotation xx+αmod1. Entre autres, nous démontrons que pour toute fonction non-polynomiale de classe C 2 admettant une série de Taylor–Fourier (i.e. les coefficients de Fourier sont nuls sur - ), il existe un α irrationnel (en réalité, un ensemble de α de deuxième catégorie au sens de Baire) tel que la série diverge pour tous les x. Nous démontrons aussi que pour tout α irrationnel, il existe une fonction continue f telle que la série diverge pour tous les x. La convergence d’une série générale n=1 a n f(x+nα) est aussi discutée pour divers cas où interviennent la propriété diophantienne du nombre α et la régularité de la fonction f.

Published online:
DOI: 10.5802/aif.3214
Classification: 37A30, 37A45
Keywords: Ergodic Hilbert transform, Everywhere divergence, Irrational rotation
Fan, Aihua 1; Schmeling, Jörg 2

1 Central China Normal University School of Mathematics and Statistics 152 Luoyu Road Wuhan 430079 (PRC)
2 Lund University Centre for Mathematical Sciences Box 118, 221 00 LUND (Sweden)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Fan, Aihua; Schmeling, Jörg. Everywhere divergence of one-sided ergodic Hilbert transform. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2477-2500. doi : 10.5802/aif.3214. https://aif.centre-mersenne.org/articles/10.5802/aif.3214/

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