On systems of dilated functions

Michel J.G. Weber

doi:10.1016/j.crma.2011.11.003

Harmonic Analysis/Dynamical Systems

On systems of dilated functions
[Sur les systèmes de fonctions dilatées]

Présenté par : Jean-Pierre Kahane

Michel J.G. Weber ¹

¹ IRMA, 7, rue René-Descartes, 67084 Strasbourg cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1261-1263.

Résumé
Abstract

Pour toute fonction $f (x) = \sum_{ℓ \in Z} a_{ℓ} e^{2 i π ℓ x}$ telle que $\sum_{ν ⩾ 1} a_{ν}^{2} Δ (ν) < \infty$ , où Δ est la fonction de Erdös–Hooley, nous montrons que la série $\sum_{k = 0}^{\infty} c_{k} f (k x)$ converge presque partout dès que la suite des coefficients vérifie

\sum_{r} (\sum_{j = 2^{r} + 1}^{2^{r + 1}} c_{j}^{2} d (j) {(\log j)}^{2})^{1 / 2} < \infty,

d (n)

désignant la fonction des diviseurs de n. Ceci améliore considérablement un certain nombre de résultats partiels précédemment obtenus.

If $f (x) = \sum_{ℓ \in Z} a_{ℓ} e^{2 i π ℓ x}$ satisfies $\sum_{ν ⩾ 1} a_{ν}^{2} Δ (ν) < \infty$ , where Δ is the Erdös–Hooley function, we show that the series $\sum_{k = 0}^{\infty} c_{k} f (k x)$ converges for almost every x, whenever the coefficient sequence verifies the condition

\sum_{r} (\sum_{j = 2^{r} + 1}^{2^{r + 1}} c_{j}^{2} d (j) {(\log j)}^{2})^{1 / 2} < \infty,

d being the divisor function. This strongly improves earlier related results.

Reçu le : 2011-05-17
Accepté le : 2011-11-04
Publié le : 2011-11-16

DOI : 10.1016/j.crma.2011.11.003

Affiliations des auteurs :

Michel J.G. Weber ¹

¹ IRMA, 7, rue René-Descartes, 67084 Strasbourg cedex, France

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Michel J.G. Weber. On systems of dilated functions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1261-1263. doi : 10.1016/j.crma.2011.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.003/

Bibliographie
Cité par

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[2] I. Berkes; M. Weber On the convergence of $\sum c_{k} f (n_{k} x)$ , Memoirs of the A.M.S., Volume 201 (2009) no. 943 (vi+72p)

[3] L. Carleson On convergence and growth of partial sums of Fourier series, Acta Math., Volume 116 (1966), pp. 135-157

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Zoltán Buczolich Almost everywhere convergence questions of series of translates of non-negative functions, Real Analysis Exchange, Volume 48 (2023) no. 1, pp. 49-76 | DOI:10.14321/realanalexch.48.1.1663223339 | Zbl:1538.28004
Michel J. G. Weber An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann zeta function, Journal of Number Theory, Volume 162 (2016), pp. 137-179 | DOI:10.1016/j.jnt.2015.10.002 | Zbl:1381.40004
Jeffrey C. Lagarias; Wen-Ching Winnie Li The Lerch zeta function. IV: Hecke operators, Research in the Mathematical Sciences, Volume 3 (2016), p. 39 (Id/No 33) | DOI:10.1186/s40687-016-0082-9 | Zbl:1411.11081
Christoph Aistleitner; István Berkes; Kristian Seip; Michel Weber Convergence of series of dilated functions and spectral norms of GCD matrices, Acta Arithmetica, Volume 168 (2015) no. 3, pp. 221-246 | DOI:10.4064/aa168-3-2 | Zbl:1339.42008
Michel J. G. Weber On convergence almost everywhere of series of dilated functions, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 353 (2015) no. 10, pp. 883-886 | DOI:10.1016/j.crma.2015.07.010 | Zbl:1348.40003
István Berkes; Michel Weber On series $\sum c_{k} f (k x)$ and Khinchin's conjecture, Israel Journal of Mathematics, Volume 201 (2014), pp. 593-609 | DOI:10.1007/s11856-014-0036-0 | Zbl:1319.40001

Cité par 6 documents. Sources : zbMATH

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