On systems of dilated functions

Weber, Michel J.G.

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

Weber, Michel J.G.¹

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

Comptes Rendus. Mathématique, Tome 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 :

Weber, Michel J.G. ¹

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

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     title = {On systems of dilated functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1261--1263},
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Weber, Michel J.G. On systems of dilated functions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1261-1263. doi : 10.1016/j.crma.2011.11.003. http://www.numdam.org/articles/10.1016/j.crma.2011.11.003/

Bibliographie
Cité par

[1] I. Berkes, On the convergence of $\sum_{n} c_{n} f (n x)$ and the Lip 1/2 class, Trans. Amer. Math. Soc. 349 (10) 4143–4158.

[2] Berkes, I.; Weber, M. 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] Carleson, L. On convergence and growth of partial sums of Fourier series, Acta Math., Volume 116 (1966), pp. 135-157

[4] Gaposhkin, V.F. On convergence and divergence systems, Mat. Zametki, Volume 4 (1968), pp. 253-260

[5] Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1979

[6] Hooley, C. A new technique and its application to the theory of numbers, Proc. London Math. Soc. (3), Volume 38 (1979), pp. 115-151

[7] Weber, M. Dynamical Systems and Processes, IRMA Lectures in Mathematics and Theoretical Physics, vol. 14, European Mathematical Society Publishing House, 2009 (xiii+759p)

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