A complete orthonormal system of divergence

Kazarian, Kazaros

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

On construit un système complet orthonormal ${\left\{{\Theta }_n\right\}}_{\mathrm{n}=1}^{\infty },{\Theta }_n\in {\mathrm{L}}_{[0,1]}^{\infty }$ 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 ${\left\{{\Theta }_n\right\}}_{\mathrm{n}=1}^{\infty },{\Theta }_n\in {\mathrm{L}}_{[0,1]}^{\infty }$ 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.

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