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Mat. Zametki, 2004, Volume 76, Issue 6, Pages 803–811 (Mi mz149)  

This article is cited in 22 scientific papers (total in 22 papers)

Problems in the Approximation of $2\pi$-Periodic Functions by Fourier Sums in the Space $L_2(2\pi)$

V. A. Abilov, F. V. Abilova


Abstract: In this paper, using the Steklov function, we introduce the modulus of continuity and define the classes of functions $W_{2,\varphi}^{r,k}$ and $W_\varphi^{r,k}$ in the spaces $L_2$ and $C$. For the class $W_{2,\varphi}^{r,k}$, we calculate the order of the Kolmogorov width and, for the class $W_\varphi^{r,k}$, we obtain an estimate of the error of a quadrature formula.

DOI: https://doi.org/10.4213/mzm149

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English version:
Mathematical Notes, 2004, 76:6, 749–757

Bibliographic databases:

UDC: 517.51.4
Received: 12.02.2003

Citation: V. A. Abilov, F. V. Abilova, “Problems in the Approximation of $2\pi$-Periodic Functions by Fourier Sums in the Space $L_2(2\pi)$”, Mat. Zametki, 76:6 (2004), 803–811; Math. Notes, 76:6 (2004), 749–757

Citation in format AMSBIB
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\paper Problems in the Approximation of $2\pi$-Periodic Functions by Fourier Sums in the Space $L_2(2\pi)$
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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. V. A. Abilov, F. V. Abilova, M. K. Kerimov, “Some remarks concerning the Fourier transform in the space $L_2(\mathbb R)$”, Comput. Math. Math. Phys., 48:6 (2008), 885–891  mathnet  crossref  zmath  isi
    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, G. A. Yusupov, “Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in $L_2$”, Siberian Math. J., 52:6 (2011), 1124–1136  mathnet  crossref  mathscinet  isi
    4. S. B. Vakarchuk, V. I. Zabutnaya, “Jackson–Stechkin Type Inequalities for Special Moduli of Continuity and Widths of Function Classes in the Space $L_2$”, Math. Notes, 92:4 (2012), 458–472  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. A. Abilov, M. K. Kerimov, “Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions”, Comput. Math. Math. Phys., 52:7 (2012), 985–991  mathnet  crossref  mathscinet  adsnasa  isi  elib  elib
    6. Vakarchuk S.B., Zabutnaya V.I., “On the Best Polynomial Approximation in the Space l (2) and Widths of Some Classes of Functions”, Ukr. Math. J., 64:8 (2013), 1168–1176  crossref  mathscinet  zmath  isi  scopus  scopus
    7. V. A. Abilov, M. V. Abilov, M. K. Kerimov, “Convergence rate estimates for spherical partial sums of double Fourier series”, Comput. Math. Math. Phys., 53:8 (2013), 1062–1069  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. M. Sh. Shabozov, K. Tukhliev, “Best Polynomial Approximations and the Widths of Function Classes in $L_{2}$”, Math. Notes, 94:6 (2013), 930–937  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. V. A. Abilov, F. V. Abilova, M. K. Kerimov, “Some new estimates of the Fourier transform in $\mathbb{L}_2(\mathbb{R})$”, Comput. Math. Math. Phys., 53:9 (2013), 1231–1238  mathnet  crossref  crossref  isi  elib  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. Vakarchuk S.B., “Jackson-Type Inequalities For the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean Nu - Widths For the Classes of Functions in the Space l (2) (R)”, Ukr. Math. J., 66:6 (2014), 827–856  crossref  mathscinet  zmath  isi  scopus  scopus
    12. Yusupov G.A., “Jackson'S-Stechkin's Inequality and the Values of Widths For Some Classes of Functions From l (2)”, Anal. Math., 40:1 (2014), 69–81  crossref  mathscinet  zmath  isi  scopus  scopus
    13. K. Tukhliev, “O priblizhenii periodicheskikh funktsii v $L_2$ i znacheniyakh poperechnikov nekotorykh klassov funktsii”, Model. i analiz inform. sistem, 22:1 (2015), 127–143  mathnet  mathscinet  elib
    14. S. B. Vakarchuk, “Generalized Smoothness Characteristics in Jackson-Type Inequalities and Widths of Classes of Functions in $L_2$”, Math. Notes, 98:4 (2015), 572–588  mathnet  crossref  crossref  mathscinet  isi  elib
    15. S. B. Vakarchuk, V. I. Zabutnaya, “Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions”, Math. Notes, 99:2 (2016), 222–242  mathnet  crossref  crossref  mathscinet  isi  elib
    16. M. K. Kerimov, E. V. Selimkhanov, “On exact estimates of the convergence rate of Fourier series for functions of one variable in the space $L_2[-\pi,\pi]$”, Comput. Math. Math. Phys., 56:5 (2016), 717–729  mathnet  crossref  crossref  isi  elib
    17. K. Tukhliev, “Nailuchshie priblizheniya i poperechniki nekotorykh klassov svertok v $L_{2}$”, Tr. IMM UrO RAN, 22, no. 4, 2016, 284–294  mathnet  crossref  mathscinet  elib
    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  scopus
    19. Akgun R., “Gadjieva'S Conjecture, K-Functionals, and Some Applications in Weighted Lebesgue Spaces”, Turk. J. Math., 42:3 (2018), 1484–1503  crossref  mathscinet  isi  scopus
    20. Akgun R., “Mixed Modulus of Smoothness With Muckenhoupt Weights and Approximation By Angle”, Complex Var. Elliptic Equ., 64:2 (2019), 330–351  crossref  mathscinet  zmath  isi  scopus
    21. M. Sh. Shabozov, A. A. Shabozova, “Nekotorye tochnye neravenstva tipa Dzheksona - Stechkina dlya periodicheskikh differentsiruemykh v smysle Veilya funktsii v $L_2$”, Tr. IMM UrO RAN, 25, no. 4, 2019, 255–264  mathnet  crossref  elib
    22. A. N. Morozov, “Vychislenie proizvodnykh v prostranstvakh $L_p$, $1 \le p \le \infty$”, Model. i analiz inform. sistem, 27:1 (2020), 124–131  mathnet  crossref
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