Everywhere divergence of one-sided ergodic Hilbert transform

Fan, Aihua; Schmeling, Jörg

doi:10.5802/aif.3214

Everywhere divergence of one-sided ergodic Hilbert transform
[Divergence partout de la transformée de Hilbert ergodique latérale]

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)

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2477-2500.

Résumé
Abstract

Etant donné un nombre $α \in (0, 1)$ et une fonction $1$ -périodique $f$ , nous étudions la convergence de la série $\sum_{n = 1}^{\infty} \frac{f (x + n α)}{n}$ , appelée la transformée de Hilbert latérale relative à la rotation $x \mapsto x + α mod 1$ . 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 $ℤ_{-}$ ), 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 $\sum_{n = 1}^{\infty} a_{n} f (x + n α)$ 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 $α \in (0, 1)$ and a $1$ -periodic function $f$ , we study the convergence of the series $\sum_{n = 1}^{\infty} \frac{f (x + n α)}{n}$ , called one-sided Hilbert transform relative to the rotation $x \mapsto x + α mod 1$ . Among others, we prove that for any non-polynomial function of class $C^{2}$ having Taylor–Fourier series (i.e. Fourier coefficients vanish on $ℤ_{-}$ ), 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 $\sum_{n = 1}^{\infty} a_{n} f (x + n α)$ 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 : 10.5802/aif.3214

Classification : 37A30, 37A45
Keywords: Ergodic Hilbert transform, Everywhere divergence, Irrational rotation
Mot clés : Transformée de Hilbert ergodique, Divergence partout, Rotation irrationelle

Affiliations des auteurs :

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)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3214},
     language = {en},
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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/articles/10.5802/aif.3214/

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