Mathematical Analysis
A complete orthonormal system of divergence
Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 85-88.

A complete orthonormal system of functions ${\left\{{\Theta }_{n}\right\}}_{\mathrm{n}=1}^{\infty },\phantom{\rule{3.30002pt}{0ex}}{\Theta }_{n}\in {\mathrm{L}}_{\left[0,1\right]}^{\infty }$ defined on the closed interval [0,1] is constructed such that ∑n=1anΘn diverges almost everywhere for any {an}n=1l2. For the constructed system the following result is true: Any nontrivial series by the system {Θn}n=1 which converges in measure to zero diverges almost everywhere.

On construit un système complet orthonormal ${\left\{{\Theta }_{n}\right\}}_{\mathrm{n}=1}^{\infty },\phantom{\rule{3.30002pt}{0ex}}{\Theta }_{n}\in {\mathrm{L}}_{\left[0,1\right]}^{\infty }$ tel que ∑n=1anΘn diverge presque partout pour n'importe quel {an}n=1l2. Pour le système construit le résultat suivant est vrai : Toute série suivant le système {Θn}n=1 non triviale et qui converge en mesure vers zéro diverge presque partout.

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

Kazaros Kazarian 1

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

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