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Izv. RAN. Ser. Mat., 1998, Volume 62, Issue 6, Pages 27–52 (Mi izv219)  

This article is cited in 10 scientific papers (total in 10 papers)

An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces

A. G. Babenko

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Let $L^2_{\alpha,\beta}$ be the Hilbert space of real-valued functions on $[0,\pi]$ with scalar product
$$ (F,G)=\int_{0}^{\pi}F(x)G(x)(\sin\dfrac{x}{2})^{2\alpha+1} (\cos\dfrac{x}{2})^{2\beta+1} dx,\qquad \alpha>-1,\quad \beta>-1, $$
and norm $\|F\|=(F,F)^{1/2}$. We prove in the case when $\alpha>\beta\geqslant-1/2$ the following exact Jackson–Stechkin inequality
$$ E_{n-1} (F)\leqslant\omega_r(F,2x_{n}^{\alpha,\beta}),\quad F\in L^2_{\alpha,\beta}, $$
between the best of $F$ by cosine-polynomials of order $n-1$ and its generalized modulus of continuity of (real) order $r\geqslant 1$: $n\geqslant\max\{2,1+ \frac{\alpha-\beta}{2}\}$ if $\beta>-\frac12$ , $n\geqslant 1$ if $\beta=-\frac12$ , where $x_{n}^{\alpha,\beta}$ is the first positive zero of the Jacobi cosine-polynomial $P_{n}^{(\alpha,\beta)}(\cos x)$. We deduce from this inequality similar inequalities for mean-square approximations of functions of several variables given on projective spaces.


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English version:
Izvestiya: Mathematics, 1998, 62:6, 1095–1119

Bibliographic databases:

MSC: 41A50, 41A10, 42A10, 41A25, 41A17
Received: 30.09.1997

Citation: A. G. Babenko, “An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces”, Izv. RAN. Ser. Mat., 62:6 (1998), 27–52; Izv. Math., 62:6 (1998), 1095–1119

Citation in format AMSBIB
\by A.~G.~Babenko
\paper An exact Jackson--Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces
\jour Izv. RAN. Ser. Mat.
\yr 1998
\vol 62
\issue 6
\pages 27--52
\jour Izv. Math.
\yr 1998
\vol 62
\issue 6
\pages 1095--1119

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    This publication is cited in the following articles:
    1. Li J., Liu Y., “The Jackson Inequality for the Best L-2-Approximation of Functions on [0,1] with the Weight x”, Numerical Mathematics-Theory Methods and Applications, 1:3 (2008), 340–356  mathscinet  zmath  isi
    2. S. S. Platonov, “Some problems in the theory of approximation of functions on compact homogeneous manifolds”, Sb. Math., 200:6 (2009), 845–885  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Ivanov V.I., Lyu Yunpin, “Otsenka snizu konstant Dzheksona v prostranstvakh $l_p$, $1\leq p<2$ s periodicheskim vesom Yakobi”, Izvestiya Tulskogo gosudarstvennogo universiteta. Seriya: Estestvennye nauki, 2011, no. 2, 59–69  elib
    4. Vo T.K., “Operatory obobschennogo sdviga v prostranstvakh $l_{p}$ na tore s vesom yakobi i ikh primenenie”, Izvestiya Tulskogo gosudarstvennogo universiteta. Seriya: Estestvennye nauki, 2012, no. 1, 17–43  elib
    5. Ivanov V.I., “Tochnye $l_2$-neravenstva dzheksona - chernykh - yudina v teorii priblizhenii”, Izvestiya tulskogo gosudarstvennogo universiteta. estestvennye nauki, 2012, no. 3, 19–28  mathscinet  elib
    6. S. S. Platonov, “Fourier–Jacobi harmonic analysis and approximation of functions”, Izv. Math., 78:1 (2014), 106–153  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. S. B. Vakarchuk, “Mean Approximation of Functions on the Real Axis by Algebraic Polynomials with Chebyshev–Hermite Weight and Widths of Function Classes”, Math. Notes, 95:5 (2014), 599–614  mathnet  crossref  crossref  mathscinet  isi  elib
    8. Vitalii Arestov, Marina Deikalova, “Nikol’skii Inequality Between the Uniform Norm and
      L q -Norm with Ultraspherical Weight of Algebraic Polynomials on an Interval”, Comput. Methods Funct. Theory, 2015  crossref  mathscinet  scopus
    9. M. Sh. Shabozov, K. Tukhliev, “Neravenstva Dzheksona — Stechkina c obobschennymi modulyami nepreryvnosti i poperechniki nekotorykh klassov funktsii”, Tr. IMM UrO RAN, 21, no. 4, 2015, 292–308  mathnet  mathscinet  elib
    10. Arestov V. Deikalova M., “Nikol'skii inequality between the uniform norm and L q -norm with Jacobi weight of algebraic polynomials on an interval”, Anal. Math., 42:2 (2016), 91–120  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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