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Mat. Zametki, 2006, Volume 80, Issue 1, Pages 11–19 (Mi mz2774)  

This article is cited in 21 scientific papers (total in 21 papers)

Jackson-Type Inequalities and Widths of Function Classes in $L_2$

S. B. Vakarchuk

Ukrainian Academy of Customs

Abstract: The sharp Jackson-type inequalities obtained by Taikov in the space $L_2$ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of $k$th order $(k=2,3,…)$. We also obtain exact values of the $n$-widths of the function classes $F(k,r,\Phi)$ and $\mathcal{F}_k^r (h)$, which are a generalization of the classes $F(1,r,\Phi)$ and $\mathcal{F}^r_1(h)$ studied by Taikov.

Keywords: Jackson-type inequalities, width of function classes, modulus of continuity of $k$th order, periodic function, Bernstein, Kolmogorov, Gelfand $n$-widths

DOI: https://doi.org/10.4213/mzm2774

Full text: PDF file (432 kB)
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English version:
Mathematical Notes, 2006, 80:1, 11–18

Bibliographic databases:

UDC: 517.5
Received: 09.03.2005
Revised: 25.12.2005

Citation: S. B. Vakarchuk, “Jackson-Type Inequalities and Widths of Function Classes in $L_2$”, Mat. Zametki, 80:1 (2006), 11–19; Math. Notes, 80:1 (2006), 11–18

Citation in format AMSBIB
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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. Shabozov M.Sh., Akobirshoev M.O., “On exact values of quasiwidths for some classes of differentiable periodic functions of two variables”, Ukrainian Math. J., 61:6 (2009), 1013–1024  crossref  mathscinet  zmath  isi  scopus
    2. S. B. Vakarchuk, V. I. Zabutnaya, “A Sharp Inequality of Jackson–Stechkin type in $L_2$ and the Widths of Functional Classes”, Math. Notes, 86:3 (2009), 306–313  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. M. Sh. Shabozov, “Widths of Classes of Periodic Differentiable Functions in the Space $L_{2}[0,2\pi]$”, Math. Notes, 87:4 (2010), 575–581  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Shabozov M.Sh., Yusupov G.A., “Inequalities between the best approximations and homogenizations of moduli of continuity in the space $L_2$”, Dokl. Math., 82:3 (2010), 892–895  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. M. Sh. Shabozov, G. A. Yusupov, “Best Polynomial Approximations in $L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths”, Math. Notes, 90:5 (2011), 748–757  mathnet  crossref  crossref  mathscinet  isi
    6. Li J. Liu Y.-p. Su Ch.-M., “The Jackson Inequality and Widths of Function Classes in l-2([0,1], X(2V+1))”, J. Complex., 28:5-6 (2012), 582–596  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Shabozov M.Sh. Yusupov G.A., “Widths of Certain Classes of Periodic Functions in l-2”, J. Approx. Theory, 164:7 (2012), 869–878  crossref  mathscinet  zmath  isi  scopus
    8. Shabozov M.Sh., “Exact Jackson-Stechkin-Type Inequalities for 2 Pi-Periodic Functions in (2) and Widths of Some Classes of Functions”, Ukr. Math. J., 63:10 (2012), 1633–1639  crossref  mathscinet  zmath  isi  scopus
    9. Yusupov G.A., “Tochnye neravenstva tipa dzheksona-stechkina i poperechniki funktsionalnykh klassov v $l_{2}$”, Izvestiya tulskogo gosudarstvennogo universiteta. estestvennye nauki, 2012, no. 2, 124–135  elib
    10. M. R. Langarshoev, “Tochnye neravenstva tipa Dzheksona–Stechkina i znacheniya poperechnikov nekotorykh klassov funktsii v prostranstve $L_{2}$”, Model. i analiz inform. sistem, 20:5 (2013), 90–105  mathnet
    11. G. A. Yusupov, “Tochnye znacheniya poperechnikov nekotorykh klassov funktsii iz $L_2$ i minimizatsiya konstant v neravenstvakh tipa Dzheksona–Stechkina”, Model. i analiz inform. sistem, 20:5 (2013), 106–116  mathnet
    12. Shabozov M.Sh. Vakarchuk S.B. Zabutnaya V.I., “Sharp Jackson-Stechkin Type Inequalities for Periodic Functions in l-2 and Widths of Function Classes”, Dokl. Math., 88:1 (2013), 478–481  crossref  mathscinet  zmath  isi  scopus
    13. 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
    14. M. R. Langarshoev, “Sharp inequality of Jackson–Stechkin type and widths of functional classes in the space $L_2$”, Eurasian Math. J., 5:1 (2014), 122–134  mathnet
    15. S. B. Vakarchuk, “Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line”, Math. Notes, 96:6 (2014), 878–896  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. Vakarchuk S.B. Shvachko A.V., “On the Best Approximation in the Mean By Algebraic Polynomials With Weight and the Exact Values of Widths For the Classes of Functions”, Ukr. Math. J., 65:12 (2014), 1774–1792  crossref  mathscinet  zmath  isi  scopus
    17. Shabozov M.Sh. Palavonov K.K., “Exact Values of Widths of Certain Classes of Periodic Differentiable Functions in the Space l (2)[0, 2 Pi]”, Anal. Math., 41:1-2 (2015), 103–115  crossref  mathscinet  zmath  isi  scopus
    18. Vakarchuk S.B., “Jackson-Type Inequalities with Generalized Modulus of Continuity and Exact Values of the n-Widths for the Classes of (?, ?)-Differentiable Functions in L 2. I”, Ukr. Math. J., 68:6 (2016), 823–848  crossref  mathscinet  isi  scopus
    19. Shabozov M.Sh. Yusupov G.A. Temurbekova S.D., J. Approx. Theory, 215 (2017), 145–162  crossref  mathscinet  zmath  isi  elib  scopus
    20. S. B. Vakarchuk, “O priblizhenii klassicheskimi ortogonalnymi polinomami s vesom v prostranstvakh $L_{2,\gamma}(a,b)$ i o poperechnikakh funktsionalnykh klassov”, Izv. vuzov. Matem., 2019, no. 12, 37–51  mathnet  crossref
    21. 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
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