A complete orthonormal system of divergence

Kazaros Kazarian

doi:10.1016/S1631-073X(03)00286-3

Mathematical Analysis

A complete orthonormal system of divergence
[Un système orthonormal complet de divergence]

Présenté par : Jean-Pierre Kahane

Kazaros Kazarian ¹

¹ Departamento de Matemáticas,C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 85-88.

Résumé
Abstract

On construit un système complet orthonormal ${Θ_{n}}_{n = 1}^{\infty}, Θ_{n} \in L_{[0, 1]}^{\infty}$ tel que ∑_n=1^∞a_nΘ_n diverge presque partout pour n'importe quel {a_n}_n=1^∞∉l². Pour le système construit le résultat suivant est vrai : Toute série suivant le système {Θ_n}_n=1^∞ non triviale et qui converge en mesure vers zéro diverge presque partout.

A complete orthonormal system of functions ${Θ_{n}}_{n = 1}^{\infty}, Θ_{n} \in L_{[0, 1]}^{\infty}$ defined on the closed interval [0,1] is constructed such that ∑_n=1^∞a_nΘ_n diverges almost everywhere for any {a_n}_n=1^∞∉l². For the constructed system the following result is true: Any nontrivial series by the system {Θ_n}_n=1^∞ which converges in measure to zero diverges almost everywhere.

Reçu le : 2003-05-30
Accepté le : 2003-06-03
Publié le : 2003-07-10

DOI : 10.1016/S1631-073X(03)00286-3

Affiliations des auteurs :

Kazaros Kazarian ¹

¹ Departamento de Matemáticas,C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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     title = {A complete orthonormal system of divergence},
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Kazaros Kazarian. A complete orthonormal system of divergence. Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 85-88. doi : 10.1016/S1631-073X(03)00286-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00286-3/

Bibliographie
Cité par

[1] N.K. Bari Trigonometric Series, A Treatise on Trigonometric Series, Pergamon Press, New York, 1961 (in Russian); English transl.:, 1964

[2] J.-P. Kahane Some Random Series of Functions, Cambridge Stud. Adv. Math., 5, Cambridge University Press, London, 1993

[3] B.S. Kashin A certain complete orthonormal system, Mat. Sb., Volume 99(141) (1976) no. 3, pp. 356-365 (in Russian); English transl.: Math. USSR-Sb., 28, 1976, pp. 315-324

[4] B.S. Kashin; A.A. Saakyan Orthogonal Series, Transl. Math. Monographs, 75, American Mathematical Society, Providence, RI, 1989

[5] K.S. Kazarian A remark on the divergence of Fourier series, Izv. Akad. Nauk Armyan. SSR Ser. Mat., Volume 18 (1983) no. 2, pp. 116-123 (in Russian); English transl.: Soviet J. Contemporary Math. Anal., 18, 1983

[6] K.S. Kazarian; D. Waterman Theorems on representations of functions by series, Mat. Sb., Volume 191 (2000) no. 12, pp. 123-140 (in Russian); English transl.: Math. USSR-Sb., 191, 11–12, 2000, pp. 1873-1889

[7] A. Khintchine; A.N. Kolmogorov Über Konvergenz von Reihen deren Glieder durch den Zufall bestimmt werden, Mat. Sb., Volume 32 (1925), pp. 668-677

[8] G. Pisier Type des espaces normés, C. R. Acad. Sci. Paris, Ser. A–B, Volume 276 (1973), pp. 1673-1676

[9] A.A. Talalyan Representation of measurable functions by series, Uspekhi Mat. Nauk, Volume 15 (1960) no. 5(95), pp. 77-141 (English transl.: Russian Math. Surveys, 15, 1960)

[10] P.L. Ulyanov Solved and unsolved problems in the theory of trigonometric and orthogonal series, Uspekhi Mat. Nauk, Volume 19 (1964) no. 1(115), pp. 3-69 (in Russian); English transl.: Russian Math. Surveys, 19, 1964

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