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Russian Mathematical Surveys, 1968, Volume 23, Issue 4, Pages 115–177
DOI: https://doi.org/10.1070/RM1968v023n04ABEH003773
(Mi rm5655)
 

This article is cited in 15 scientific papers (total in 15 papers)

Direct and converse theorems of approximation theory and semigroups of operators

N. P. Kuptsov
References:
Abstract: One of the important aims of the modem constructive theory of functions is to establish relationships between the structural properties 6i functions and sequences of approximations to them. The foundations of work in this field were laid by Jackson, Bernstein, and de la Vallée-Poussin. Subsequent developments were made by Zygmund, Kolmogorov, Nikol'skii, Pavard, and others.
Jackson' s classical inequality and the fundamental converse theorem of Bernstein-de la Vallée–Poussin, which were initially established for approximations to continuous functions by algebraic and trigonometric polynomials, have been generalized in various directions. Direct and converse theorems have been obtained for algebraic and trigonometric approximations in spaces other than C, for spaces of almost periodic functions, for approximations by eigenfunctions of a Sturm–Liouville problem, and so on.
The purpose of the present paper is to set forth the basic direct and converse theorems of the theory of approximations in Banach spaces. The main technique of the investigation is the use of strongly continuous semigroups of operators and the resolvents of operators generating these semigroups. Under certain conditions on the resolvent (see Ch. II, § 1), general direct and converse theorems are established for approximations by eigen-subspaces of a generating operator. These general theorems include as special cases many of the previously known results in the constructive theory of functions.
Received: 09.01.1968
Bibliographic databases:
Document Type: Article
UDC: 517.4+519.4
MSC: 41A30, 47D03, 46S30
Language: English
Original paper language: Russian
Citation: N. P. Kuptsov, “Direct and converse theorems of approximation theory and semigroups of operators”, Russian Math. Surveys, 23:4 (1968), 115–177
Citation in format AMSBIB
\Bibitem{Kup68}
\by N.~P.~Kuptsov
\paper Direct and converse theorems of approximation theory and semigroups of operators
\jour Russian Math. Surveys
\yr 1968
\vol 23
\issue 4
\pages 115--177
\mathnet{http://mi.mathnet.ru/eng/rm5655}
\crossref{https://doi.org/10.1070/RM1968v023n04ABEH003773}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=234189}
\zmath{https://zbmath.org/?q=an:0159.43404|0184.16501}
Linking options:
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  • https://doi.org/10.1070/RM1968v023n04ABEH003773
  • https://www.mathnet.ru/eng/rm/v23/i4/p117
  • This publication is cited in the following 15 articles:
    1. Rúben Sousa, Manuel Guerra, Semyon Yakubovich, “A unified construction of product formulas and convolutions for Sturm–Liouville operators”, Anal.Math.Phys., 11:2 (2021)  crossref
    2. S. I. Bezkryla, O. N. Nesterenko, A. V. Chaikovs'kyi, “One Inequality for the Moduli of Continuity of Fractional Order Generated by Semigroups of Operators”, Ukr Math J, 71:3 (2019), 352  crossref
    3. A. Baskakov, I. Strukova, “Harmonic analysis of functions periodic at infinity”, Eurasian Math. J., 7:4 (2016), 9–29  mathnet
    4. I. I. Strukova, “O garmonicheskom analize periodicheskikh na beskonechnosti funktsii”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 14:1 (2014), 28–38  mathnet  crossref  elib
    5. V. P. Sklyarov, “Ob uslovii s-regulyarnosti N. P. Kuptsova”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 13:1(2) (2013), 84–87  mathnet  crossref
    6. B. F. Ivanov, “Ob odnom obobschenii neravenstvo Bora”, Probl. anal. Issues Anal., 2(20):2 (2013), 21–58  mathnet  mathscinet  zmath
    7. S. A. Kreis, “Freimy i periodicheskie gruppy operatorov”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 12:2 (2012), 14–18  mathnet  crossref  elib
    8. G. V. Radzievskii, “Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree”, Sb. Math., 192:12 (2001), 1829–1876  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. P. K. Suetin, B. I. Golubov, A. F. Leont'ev, M. I. Voǐtsekhovskiǐ, S. A. Aǐvazyan, A. Shtern, L. V. Kuz'min, A. A. Sapozhenko, K. A. Borovkov, M. S. Nikulin, V. P. Maslov, P. S. Modenov, A. I. Shtern, A. G. Dragalin, Vik. S. Kulikov, V. I. Nechaev, E. P. Dolzhenko, E. D. Solomentsev, T. P. Lukashenko, Yu. N. Subbotin, L. D. Ivanov, A. V. Arkhangel'skiǐ, V. I. Ponomarev, E. B. Vinberg, S. A. Telyakovskiǐ, I. I. Volkov, S. N. Smirnov, A. V. Tolstikov, S. A. Stepanov, V. M. Babich, D. D. Sokolov, L. D. Kudryavtsev, D. N. Zubarev, I. V. Proskuryakov, R. A. Minlos, Yu. P. Ivanilov, V. V. Okhrimenko, N. N. Vorob'ev, B. A. Pasynkov, M. Sh. Tsalenko, A. D. Kuz'min, B. L. Laptev, V. S. Malakhovskiǐ, V. I. Malykhin, T. S. Fofanova, A. L. Onishchik, V. E. Plisko, V. N. Latyshev, A. I. Kostrikin, I. V. Dolgachev, Yu. I. Yanov, Yu. I. Merzlyakov, O. A. Ivanova, A. N. Parshin, S. N. Artemov, G. S. Asanov, A. D. Aleksandrov, V. N. Berestovsk, Encyclopaedia of Mathematics, 1995, 549  crossref
    10. I. A. Vinogradova, A. G. El'kin, Yu. V. Prokhorov, B. A. Efimov, L. P. Kuptsov, N. Kh. Rozov, V. A. Oskolkov, L. D. Kudryavtsev, B. V. Khvedelidze, A. A. Zakharov, M. Sh. Tsalenko, E. D. Solomentsev, Yu. L. Ershov, I. V. Dolgachev, B. B. Venkov, A. N. Parshin, A. I. Kostrikin, A. B. Ivanov, A. P. Terekhin, V. F. Emelyanov, V. V. Sazonov, M. I. Voǐtsekhovskiǐ, I. I. Volkov, P. S. Aleksandrov, A. V. Prokhorov, A. M. Zubkov, V. N. Grishin, A. A. Danilevich, N. M. Nagornyǐ, E. G. D'yakonov, Kh. D. Ikramov, N. S. Bakhvalov, A. V. Arkhangel'skiǐ, V. V. Rumyantsev, A. V. Zarelua, A. A. Mal'tsev, O. A. Ivanova, V. P. Fedotov, I. P. Kubilyus, B. M. Bredikhin, P. L. Dobrushin, V. V. Prelov, A. V. Mikhalev, V. A. Andrunakievich, V. V. Fedorchuk, V. P. Platonov, A. P. Favorskiǐ, D. V. Anosov, V. I. Danilov, E. L. Tonkov, A. L. Onishchik, T. S. Pigolkina, T. S. Pogolkina, L. A. Skornyakov, V. I. Sobolev, I. Kh. Sabitov, V. I. Lebedev, A. V. Lykov, A., Encyclopaedia of Mathematics, 1995, 1  crossref
    11. A. G. Baskakov, “Spectral analysis of perturbed nonquasianalytic and spectral operators”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 1–31  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. M. Hazewinkel, Encyclopaedia of Mathematics, 1988, 1  crossref
    13. A. P. Terekhin, “A multiparameter semigroup of operators, mixed moduli and aproximation”, Math. USSR-Izv., 9:4 (1975), 887–910  mathnet  crossref  mathscinet  zmath
    14. K. K. Golovkin, “Uniform equivalence of parametric norms in ergodic and approximation theories”, Math. USSR-Izv., 5:4 (1971), 915–934  mathnet  crossref  mathscinet  zmath
    15. Pure and Applied Mathematics, 40, 1971, 521  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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