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Mat. Zametki, 2011, Volume 90, Issue 5, Pages 764–775 (Mi mz8821)  

This article is cited in 23 scientific papers (total in 23 papers)

Best Polynomial Approximations in $L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths

M. Sh. Shabozova, G. A. Yusupovb

a Institute of Mathematics, Academy of Sciences of Republic of Tajikistan
b Tajik National University, Dushanbe

Abstract: We consider the problem of determining sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of $m$th order in the space $L_2$ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of $n$-widths in $L_2$.

Keywords: best polynomial approximation, periodic differentiable function, trigonometric polynomial, modulus of continuity, the space $L_2$, $n$-width, Fourier series

DOI: https://doi.org/10.4213/mzm8821

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English version:
Mathematical Notes, 2011, 90:5, 748–757

Bibliographic databases:

UDC: 517.5
Received: 22.02.2010
Revised: 29.09.2010

Citation: 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”, Mat. Zametki, 90:5 (2011), 764–775; Math. Notes, 90:5 (2011), 748–757

Citation in format AMSBIB
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\paper Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths
\jour Mat. Zametki
\yr 2011
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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. Yusupov G.A., “Tochnye neravenstva tipa Dzheksona-Stechkina i poperechniki funktsionalnykh klassov v $l_{2}$”, Izv. Tulskogo gos. un-ta. Estestvennye nauki, 2012, no. 2, 124–135  elib
    2. Shabozov M.Sh., Temurbekova S.D., “Znacheniya poperechnikov klassov funktsii iz $l_{2}[0,2\pi]$ i minimizatsiya tochnykh konstant v neravenstvakh tipa Dzheksona”, Izv. Tulskogo gos. un-ta. Estestvennye nauki, 2012, no. 3, 60–68  elib
    3. 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
    4. 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
    5. Langarshoev M.R., “Nailuchshie priblizheniya nekotorykh klassov periodicheskikh funktsii v $L_2[0,2\pi]$”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:2 (2013), 100–106  elib
    6. Yusupov G.A., “O nailuchshikh srednekvadraticheskikh priblizheniyakh na vsei osi tselymi funktsiyami eksponentsialnogo tipa”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:3 (2013), 192–195  elib
    7. Tukhliev K., “O nailuchshem polinomialnom priblizhenii periodicheskikh funktsii v $L_2$ i poperechnikov nekotorykh klassov funktsii”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:7 (2013), 515–520  elib
    8. Tukhliev K., “O nailuchshikh priblizheniyakh tselymi funktsiyami v prostranstve. I”, Izvestiya Akademii nauk Respubliki Tadzhikistan. Otdelenie fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2013, no. 3(152), 19–29  elib
    9. Shabozov M.Sh., Oliftaev N.F., “Nailuchshie priblizheniya i tochnye znacheniya poperechnikov nekotorykh klassov periodicheskikh funktsii v $L_2$”, Izvestiya Akademii nauk Respubliki Tadzhikistan. Otdelenie fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2013, no. 4(153), 23–32  elib
    10. Palavonov K.K., “O nailuchshem priblizhenii periodicheskikh funktsii i znacheniyakh poperechnikov funktsionalnykh klassov v $L_2$”, Izvestiya Akademii nauk Respubliki Tadzhikistan. Otdelenie fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2013, no. 2(151), 40–51  mathscinet  elib
    11. Shabozov M.Sh., Langarshoev M.R., “O nailuchshikh priblizheniyakh i tochnykh znacheniyakh srednikh poperechnikov nekotorykh klassov tselykh funktsii v $L_2(\mathbb R)$”, Izvestiya Akademii nauk Respubliki Tadzhikistan. Otdelenie fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2013, no. 1(150), 7–20  elib
    12. 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
    13. 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
    14. 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
    15. 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
    16. M. Sh. Shabozov, A. D. Farozova, “Tochnoe neravenstvo Dzheksona–Stechkina s neklassicheskim modulem nepreryvnosti”, Tr. IMM UrO RAN, 22, no. 4, 2016, 311–319  mathnet  crossref  mathscinet  elib
    17. K. Tukhliev, “Srednekvadraticheskoe priblizhenie funktsii ryadami Fure–Besselya i znacheniya poperechnikov nekotorykh funktsionalnykh klassov”, Chebyshevskii sb., 17:4 (2016), 141–156  mathnet  crossref  elib
    18. S. B. Vakarchuk, “Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(\psi, \beta)$-differentiable functions in $L_2$. II”, Ukrainian Math. J., 68:8 (2017), 1165–1183  crossref  mathscinet  isi  elib  scopus
    19. S. B. Vakarchuk, “Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(\psi, \beta)$-differentiable functions in $L_2$. I”, Ukrainian Math. J., 68:6 (2016), 823–848  crossref  mathscinet  isi  elib  scopus
    20. Shabozov M.Sh. Yusupov G.A. Temurbekova S.D., “$N$-widths of certain function classes defined by the modulus of continuity”, J. Approx. Theory, 215 (2017), 145–162  crossref  mathscinet  zmath  isi  elib  scopus
    21. 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  isi  elib
    22. M. S. Saidusainov, “Analiz odnoi teoremy o neravenstve Dzheksona - Stechkina v prostranstve Bergmana $B_2$”, Tr. IMM UrO RAN, 24, no. 4, 2018, 217–224  mathnet  crossref  elib
    23. 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
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