The explicit characterization of coefficients of a.e. convergent orthogonal series

Adam Paszkiewicz

doi:10.1016/j.crma.2009.07.012

Probability Theory/Mathematical Analysis

The explicit characterization of coefficients of a.e. convergent orthogonal series
[Caractérisation explicite des coefficients des séries orthogonales convergentes presques partout]

Présenté par : Michel Talagrand

Adam Paszkiewicz ¹

¹ Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, PL-90-238 Łódź, Poland

Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1213-1216.

Résumé
Abstract

On donne une complète caractérisation de la suite des nombres $(a_{n})$ telle que $\sum_{n ⩾ 1} a_{n} Φ_{n}$ converge, presque partout, pour tout système orthogonal $(Φ_{n})$ dans tout espace $L_{2}$ .

La démonstration détaillées est donnée par A. Paszkiewicz dans l'article : On complete characterization of coefficients of a.e. convergent orthogonal series.

We characterize sequences of numbers $(a_{n})$ such that $\sum_{n ⩾ 1} a_{n} Φ_{n}$ converges a.e. for any orthonormal system $(Φ_{n})$ in any $L_{2}$ -space.

Reçu le : 2008-03-16
Accepté le : 2009-07-15
Publié le : 2009-09-15

DOI : 10.1016/j.crma.2009.07.012

Affiliations des auteurs :

Adam Paszkiewicz ¹

¹ Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, PL-90-238 Łódź, Poland

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     title = {The explicit characterization of coefficients of a.e. convergent orthogonal series},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1213--1216},
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     year = {2009},
     doi = {10.1016/j.crma.2009.07.012},
     language = {en},
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Adam Paszkiewicz. The explicit characterization of coefficients of a.e. convergent orthogonal series. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1213-1216. doi : 10.1016/j.crma.2009.07.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.012/

Bibliographie
Cité par

[1] V.F. Gaposhkin On convergence of orthogonal series, Dokl. Akad. Nauk SSSR, Volume 159 (1964), pp. 243-246 (in Russian)

[2] A. Paszkiewicz, On complete characterization of coefficients of a.e. convergent orthogonal series and on majorizing measures, Invent. Math., in press

[3] M. Talagrand Sample boundedness of stochastic processes under increment conditions, Ann. Probab., Volume 18 (1990), pp. 1-49

[4] M. Talagrand, Convergence of orthogonal series using stochastic processes, unpublished manuscript

[5] K. Tandori Über die Konvergenz der Orthogonalreihen, Acta Sci. Math. (Szeged), Volume 24 (1963), pp. 139-151

[6] M. Weber Some theorems related to almost sure convergence of orthogonal series, Indag. Math. (N.S.), Volume 11 (2000), pp. 293-311

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