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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 1, Pages 117–166 (Mi izv7998)  

This article is cited in 3 scientific papers (total in 3 papers)

Fourier–Jacobi harmonic analysis and approximation of functions

S. S. Platonov

Petrozavodsk State University

Abstract: We use the methods of Fourier–Jacobi harmonic analysis to study problems of the approximation of functions by algebraic polynomials in weighted function spaces on $[-1,1]$. We prove analogues of Jackson's direct theorem for the moduli of smoothness of all orders constructed on the basis of Jacobi generalized translations. The moduli of smoothness are shown to be equivalent to $K$-functionals constructed from Sobolev-type spaces. We define Nikol'skii–Besov spaces for the Jacobi generalized translation and describe them in terms of best approximations. We also prove analogues of some inverse theorems of Stechkin.

Keywords: Fourier–Jacobi harmonic analysis, approximation of functions, generalized translations, Jacobi polynomials, function spaces.

DOI: https://doi.org/10.4213/im7998

Full text: PDF file (808 kB)
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English version:
Izvestiya: Mathematics, 2014, 78:1, 106–153

Bibliographic databases:

UDC: 517.518.8
MSC: 41A10, 42A10, 42C05, 33C45
Received: 10.05.2012
Revised: 10.11.2012

Citation: S. S. Platonov, “Fourier–Jacobi harmonic analysis and approximation of functions”, Izv. RAN. Ser. Mat., 78:1 (2014), 117–166; Izv. Math., 78:1 (2014), 106–153

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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. Y. Kolomoitsev, T. Lomako, J. Prestin, “On approximation of functions by algebraic polynomials in Holder spaces”, Mathematische Nachrichten, 289:16 (2016), 2037–2057  crossref  mathscinet  zmath  isi  elib  scopus
    2. S. El Ouadih, R. Daher, “Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on $[0, \pi]$”, C. R. Math. Acad. Sci. Paris, 355:3 (2017), 318–324  crossref  mathscinet  zmath  isi  scopus
    3. Platonov S.S., “Fourier-Jacobi Harmonic Analysis and Some Problems of Approximation of Functions on the Half-Axis in l-2 Metric: Jackson'S Type Direct Theorems”, Integral Transform. Spec. Funct., 30:4 (2019), 264–281  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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