Geodesic
On approximation properties of certain incomplete systems
Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 443-471 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\{\varphi_n(x)\}$ be a system of almost-everywhere finite measurable functions on $[0,1]$ that has one of the following properties: I. $\{\varphi_n(x)\}^\infty_{n=1}$ is a system for representing the functions in $L_p[0,1]$, $0, by convergent series. II. $\{\varphi_n(x)\}^\infty_{n=1}$ is a system for representing the functions in $L_p[0,1]$, $0, by almost-everywhere convergent series. III. $\{\varphi_n(x)\}^\infty_{n=1}$ has the strong Luzing $C$-property. IV. $\{\varphi_n(x)\}^\infty_{n=1}$ can be multiplicatively completed to form a system for representing the functions in $L_p[0,1]$, $p\geqslant1$, by series that converge in the $L_p[0,1]$-metric. It is shown that if $\{\varphi_n(x)\}^\infty_{n=1}$ is a system having one of the properties I–IV, then any subsystem of it with the form $\{\varphi_k(x)\}^\infty_{k=N+1}$ ($N$ any natural number) also has this property. Bibliography: 9 titles. 
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A. A. Talalyan. On approximation properties of certain incomplete systems. Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 443-471. http://geodesic.mathdoc.fr/item/SM_1982_43_4_a0/

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