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 Mat. Sb. (N.S.), 1981, Volume 115(157), Number 4(8), Pages 499–531 (Mi msb2412)

On approximation properties of certain incomplete systems

A. A. Talalyan

Abstract: 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 $L_p[0,1]$, $0<p<1$, by convergent series.
II. $\{\varphi_n(x)\}^\infty_{n=1}$ is a system for representing the functions in $L_p[0,1]$, $0<p<1$, 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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English version:
Mathematics of the USSR-Sbornik, 1982, 43:4, 443–471

Bibliographic databases:

UDC: 517.52
MSC: Primary 42C15, 46E30; Secondary 46B15

Citation: A. A. Talalyan, “On approximation properties of certain incomplete systems”, Mat. Sb. (N.S.), 115(157):4(8) (1981), 499–531; Math. USSR-Sb., 43:4 (1982), 443–471

Citation in format AMSBIB
\Bibitem{Tal81} \by A.~A.~Talalyan \paper On~approximation properties of certain incomplete systems \jour Mat. Sb. (N.S.) \yr 1981 \vol 115(157) \issue 4(8) \pages 499--531 \mathnet{http://mi.mathnet.ru/msb2412} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=629624} \zmath{https://zbmath.org/?q=an:0503.42025} \transl \jour Math. USSR-Sb. \yr 1982 \vol 43 \issue 4 \pages 443--471 \crossref{https://doi.org/10.1070/SM1982v043n04ABEH002574} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Ivanov V., “Representation of Functions by Series in Metric Symmetrical-Spaces Without Linear Functionals”, 289, no. 3, 1986, 532–535
2. A. A. Talalyan, R. I. Ovsepian, “The representation theorems of D. E. Men'shov and their impact on the development of the metric theory of functions”, Russian Math. Surveys, 47:5 (1992), 13–47
3. V. I. Filippov, “Function systems obtained using translates and dilates of a single function in the paces $E_\varphi$ with $\lim_{t\to\infty}\frac{\varphi(t)}t=0$”, Izv. Math., 65:2 (2001), 389–402
4. Filippov V.I., “Sistemy szhatii i sdvigov odnoi funktsii v mnogomernykh prostranstvakh e”, Vestnik saratovskogo gosudarstvennogo sotsialno-ekonomicheskogo universiteta, 2011, no. 1, 120–122
5. V. I. Filippov, “Representation systems obtained using translates and dilates of a single function in multidimensional spaces $E_{\varphi}$”, Izv. Math., 76:6 (2012), 1257–1270
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