On the convergence of orthogonal series

Bednorz, Witold

doi:10.1016/j.crma.2011.02.001

Probability Theory

On the convergence of orthogonal series
[Sur la convergence des systèmes orthogonaux]

Présenté par : Michel Talagrand

Bednorz, Witold ¹

¹ Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 455-458.

Résumé
Abstract

Nous proposons une nouvelle approche pour démontrer que la convergence presque sure de la série $\sum_{n = 1}^{\infty} a_{n} φ_{n}$ pour tous les systèmes orthogonaux ${(φ_{n})}_{n = 1}^{\infty}$ est équivalente à lʼexistence dʼune mesure majorante sur lʼensemble $T = {\sum_{m = n}^{\infty} a_{m}^{2} : n \geq 1} \cup {0}$ . Lʼingrédient principal est une nouvelle méthode de construction de séries orthogonales.

In this Note we present a new approach to the complete characterization of the a.s. convergence of orthogonal series. We sketch a new proof that a.s. convergence of $\sum_{n = 1}^{\infty} a_{n} φ_{n}$ for all orthonormal systems ${(φ_{n})}_{n = 1}^{\infty}$ is equivalent to the existence of a majorizing measure on the set $T = {\sum_{m = n}^{\infty} a_{m}^{2} : n \geq 1} \cup {0}$ . The method is based on the chaining argument used for a certain partitioning scheme.

Reçu le : 2011-02-01
Accepté le : 2011-02-04
Publié le : 2011-03-04

DOI : 10.1016/j.crma.2011.02.001

Affiliations des auteurs :

Bednorz, Witold ¹

¹ Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

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Bednorz, Witold. On the convergence of orthogonal series. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 455-458. doi : 10.1016/j.crma.2011.02.001. http://www.numdam.org/articles/10.1016/j.crma.2011.02.001/

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