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Mat. Zametki, 2014, Volume 95, Issue 5, Pages 666–684 (Mi mz9299)  

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

Mean Approximation of Functions on the Real Axis by Algebraic Polynomials with Chebyshev–Hermite Weight and Widths of Function Classes

S. B. Vakarchuk

Alfred Nobel University Dnepropetrovsk

Abstract: We obtain sharp Jackson–Stechkin type inequalities on the sets $L^r_{2,\rho}(\mathbb{R})$ in which the values of best polynomial approximations are estimated from above via both the moduli of continuity of $m$th order and $K$-functionals of $r$th derivatives. For function classes defined by these characteristics, the exact values of various widths are calculated in the space $L_{2,\rho}(\mathbb{R})$. Also, for the classes $W^r_{2,\rho}(\mathbb{K}_m,\Psi)$, where $r=2,3,…$, the exact values of the best polynomial approximations of the intermediate derivatives $f^{(\nu)}$, $\nu=1,…,r-1$, are obtained in $L_{2,\rho}(\mathbb{R})$.

Keywords: mean approximation by algebraic polynomials, Jackson–Stechkin type inequalities, Chebyshev–Hermite weight, width of a function class, Fourier–Hermite series, modulus of continuity, Hölder's inequality.

DOI: https://doi.org/10.4213/mzm9299

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English version:
Mathematical Notes, 2014, 95:5, 599–614

Bibliographic databases:

UDC: 517.5
Received: 22.12.2011
Revised: 23.03.2013

Citation: S. B. Vakarchuk, “Mean Approximation of Functions on the Real Axis by Algebraic Polynomials with Chebyshev–Hermite Weight and Widths of Function Classes”, Mat. Zametki, 95:5 (2014), 666–684; Math. Notes, 95:5 (2014), 599–614

Citation in format AMSBIB
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\pages 666--684
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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. Mukim S. Saidusajnov, “$\mathcal{K}$-functionals and exact values of $n$-widths in the Bergman space”, Ural Math. J., 3:2 (2017), 74–81  mathnet  crossref  mathscinet
    2. M. Sh. Shabozov, M. S. Saidusajnov, “Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space $L_2$ and $n$-Widths”, Math. Notes, 103:4 (2018), 656–668  mathnet  crossref  crossref  mathscinet  isi  elib
    3. S. B. Vakarchuk, “Approximation by classical orthogonal polynomials with weight in spaces $L_{2,\gamma}(a,b)$ and widths of some functional classes”, Russian Math. (Iz. VUZ), 63:12 (2019), 32–44  mathnet  crossref  crossref  isi
    4. M. Sh. Shabozov, M. S. Saidusainov, “Srednekvadraticheskoe priblizhenie funktsii kompleksnogo peremennogo summami Fure po ortogonalnym sistemam”, Tr. IMM UrO RAN, 25, no. 2, 2019, 258–272  mathnet  crossref  elib
    5. S. B. Vakarchuk, “On the estimates of widths of the classes of functions defined by the generalized moduli of continuity and majorants in the weighted space l-2,l-x(0,1)”, Ukr. Math. J., 71:2 (2019), 202–214  crossref  mathscinet  isi
    6. M. Sh. Shabozov, O. A. Dzhurakhonov, “Upper bounds for the approximation of some classes of bivariate functions by triangular Fourier-Hermite sums in the space l-2,l-rho(double-struck capital R-2)”, Anal. Math., 45:4 (2019), 823–840  crossref  mathscinet  isi
    7. M. Sh. Shabozov, Kh. M. Khuromonov, “On the best approximation in the mean of functions of a complex variable by Fourier series in the Bergman space”, Russian Math. (Iz. VUZ), 64:2 (2020), 66–83  mathnet  crossref  crossref  isi
    8. K. Tukhliev, A. M. Tuichiev, “Srednekvadraticheskoe priblizhenie funktsii na vsei osi s vesom Chebysheva - Ermita algebraicheskimi polinomami”, Tr. IMM UrO RAN, 26, no. 2, 2020, 270–277  mathnet  crossref  elib
    9. O. A. Dzhurakhonov, “Priblizhenie funktsii dvukh peremennykh «krugovymi» summami Fure — Chebysheva v $L_{2,\rho}$”, Vladikavk. matem. zhurn., 22:2 (2020), 5–17  mathnet  crossref
    10. M. Sh. Shabozov, O. A. Dzhurakhonov, “Priblizhenie v srednem nekotorykh klassov funktsii dvukh peremennykh summami Fure - Chebysheva”, Tr. IMM UrO RAN, 26, no. 4, 2020, 268–278  mathnet  crossref  elib
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