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

This article is cited in 5 scientific papers (total in 5 papers)

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
Received: 29.12.1980

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
    Related articles on Google Scholar: Russian articles, English articles

    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  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath
    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  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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