Everywhere divergence of one-sided ergodic Hilbert transform  [ Divergence partout de la transformée de Hilbert ergodique latérale ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2477-2500.

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.

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.

Reçu le : 2016-10-01
Révisé le : 2017-07-21
Accepté le : 2017-12-13
Publié le : 2018-11-23
DOI : https://doi.org/10.5802/aif.3214
Classification : 37A30,  37A45
Mots clés: Transformée de Hilbert ergodique, Divergence partout, Rotation irrationelle
@article{AIF_2018__68_6_2477_0,
     author = {Fan, Aihua and Schmeling, J\"org},
     title = {Everywhere divergence of one-sided ergodic Hilbert transform},
     journal = {Annales de l'Institut Fourier},
     pages = {2477--2500},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3214},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2018__68_6_2477_0/}
}
Fan, Aihua; Schmeling, Jörg. Everywhere divergence of one-sided ergodic Hilbert transform. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2477-2500. doi : 10.5802/aif.3214. https://aif.centre-mersenne.org/item/AIF_2018__68_6_2477_0/

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