Comptes Rendus
Harmonic Analysis/Dynamical Systems
On systems of dilated functions
[Sur les systèmes de fonctions dilatées]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1261-1263.

Pour toute fonction f(x)=Zae2iπx telle que ν1aν2Δ(ν)<, où Δ est la fonction de Erdös–Hooley, nous montrons que la série k=0ckf(kx) converge presque partout dès que la suite des coefficients vérifie

r(j=2r+12r+1cj2d(j)(logj)2)1/2<,
d(n) désignant la fonction des diviseurs de n. Ceci améliore considérablement un certain nombre de résultats partiels précédemment obtenus.

If f(x)=Zae2iπx satisfies ν1aν2Δ(ν)<, where Δ is the Erdös–Hooley function, we show that the series k=0ckf(kx) converges for almost every x, whenever the coefficient sequence verifies the condition

r(j=2r+12r+1cj2d(j)(logj)2)1/2<,
d being the divisor function. This strongly improves earlier related results.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.11.003

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/

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

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

[7] M. Weber 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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