On the convergence of orthogonal series

Witold Bednorz

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

Witold Bednorz ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Witold Bednorz ¹

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

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     title = {On the convergence of orthogonal series},
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     year = {2011},
     doi = {10.1016/j.crma.2011.02.001},
     language = {en},
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Witold Bednorz. On the convergence of orthogonal series. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 455-458. doi : 10.1016/j.crma.2011.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.001/

Bibliographie
Cité par

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[2] W. Bednorz Majorizing measures on metric spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2009) no. 1–2, pp. 75-78

[3] X. Fernique Régularité de fonctions aléatoires non gaussiennes, Ecole dʼÉté de Probabilités de Saint-Flour XI-1981, Lecture Notes in Mathematics, vol. 976, Springer, 1983, pp. 1-74

[4] B.S. Kashin; A.A. Saakyan Orthogonal Series, Translation of Mathematical Monographs, vol. 75, Amer. Math. Soc., 1989

[5] R. Latala Sudakov minoration principle and supremum of some processes, Geom. Funct. Anal., Volume 7 (1997), pp. 936-953

[6] F. Moricz; K. Tandori An improved Menchov–Rademacher theorem, Proc. Amer. Math. Soc., Volume 124 (1996), pp. 877-885

[7] A. Paszkiewicz The explicit characterization of coefficients of a.e. convergent orthogonal series, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2008) no. 19–20, pp. 1213-1216

[8] A. Paszkiewicz A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures, Invent. Math., Volume 180 (2010), pp. 55-110

[9] M. Talagrand Regularity of Gaussian processes, Acta Math., Volume 159 (1987) no. 1–2, pp. 99-149

[10] M. Talagrand Sample boundedness of stochastic processes under increment conditions, Ann. Probab., Volume 18 (1990) no. 1, pp. 1-49

[11] M. Talagrand The supremum of some canonical processes, Amer. J. Math., Volume 116 (1994) no. 2, pp. 283-325

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

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