Geodesic
On completeness of a system of functions that are close to power functions
Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 400-418 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain necessary and sufficient conditions for the completeness of the system $\{f_n(x)=x^{\lambda_n}[1+\varepsilon_n(x)]\}$ in $L_p[0, a]$, where $\varepsilon_n(x)\in L_p[0, a]$, $\varlimsup_{n\to\infty}\frac{\ln m_n}{\lambda_n}<0$, $m_n = \|\varepsilon_n(x)\|_{L_p[0, a]}$, $n=1,2,\dots$; $1. Bibliography: 4 titles. 
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L. A. Leont'eva. On completeness of a system of functions that are close to power functions. Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 400-418. http://geodesic.mathdoc.fr/item/SM_1971_13_3_a4/

[1] N. I. Akhiezer, Lektsii po teorii approksimatsii, Nauka, Moskva, 1965 | MR

[2] L. A. Leonteva, “O polnote odnoi sistemy funktsii na otrezke”, Matem. zametki, 4:5 (1968), 557–568 | MR

[3] R. Boas, Entire functions, Academic Press Inc., New York, 1954 | MR | Zbl

[4] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956