Convergence of trigonometric series with general monotone coefficients

Dyachenko, Mikhail; Tikhonov, Sergey

doi:10.1016/j.crma.2007.06.009

Mathematical Analysis

Convergence of trigonometric series with general monotone coefficients
[Convergence de séries trigonométriques à coefficients généraux monotones]

Présenté par : Jean-Pierre Kahane

Dyachenko, Mikhail ¹ ; Tikhonov, Sergey ²

¹ Moscow State University, Vorobevy Gory, 117234 Moscow, Russia
² Scuola Normale Superiore, Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Comptes Rendus. Mathématique, Tome 345 (2007) no. 3, pp. 123-126.

Résumé
Abstract

Dans cette Note on établit des résultats de convergence de séries trigonométriques dans des espaces $L_{p}$ unidimensionnels et sur le tore à n-dimensions. Des conditions suffisantes et des critères de validité sont établis pour des séries à coefficients généraux monotones. Un théorème de type Hardy–Littlewood est démontré dans le cas de séries multidimensionnelles. Des corollaires sont énoncés, en particulier dans le cas des séries multidimensionnelles.

In this Note we study the convergence results for trigonometric series in $L_{p}$ -spaces on one-dimensional and n-dimension torus. The sufficient conditions for these results to hold as well as criteria are written for the series with general monotone coefficients. The Hardy–Littlewood type theorem is obtained for multiple series. Several corollaries, in particular, u-convergence are presented.

Reçu le : 2007-02-13
Accepté le : 2007-06-05
Publié le : 2007-07-13

DOI : 10.1016/j.crma.2007.06.009

Affiliations des auteurs :

Dyachenko, Mikhail ¹ ; Tikhonov, Sergey ²

¹ Moscow State University, Vorobevy Gory, 117234 Moscow, Russia
² Scuola Normale Superiore, Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

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Dyachenko, Mikhail; Tikhonov, Sergey. Convergence of trigonometric series with general monotone coefficients. Comptes Rendus. Mathématique, Tome 345 (2007) no. 3, pp. 123-126. doi : 10.1016/j.crma.2007.06.009. http://www.numdam.org/articles/10.1016/j.crma.2007.06.009/

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