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Mat. Zametki, 2010, Volume 87, Issue 4, Pages 616–623 (Mi mz7707)  

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

Widths of Classes of Periodic Differentiable Functions in the Space $L_{2}[0,2\pi]$

M. Sh. Shabozov

Institute of Mathematics, Academy of Sciences Republic of Tajikistan

Abstract: We obtain exact values of different $n$-widths for classes of differentiable periodic functions in the space $L_{2}[0,2\pi]$ satisfying the constraint
$$ (\int_{0}^{h}\omega_{m}^{p}(f^{(r)};t) dt)^{1/p}\le\Phi(h), $$
where $0<h<\infty$, $1/r<p\le2$, $r\in\mathbb{N}$, and $\omega_{m}(f^{(r)};t)$ is the modulus of continuity of $m$th order of the derivative $f^{(r)}(x)\in L_{2}[0,2\pi]$.

Keywords: differentiable periodic function, width in the sense of Bernstein, Kolmogorov, Gelfand, the space $L_{2}[0,2\pi]$, trigonometric polynomial, Fourier series, modulus of continuity, linear operator

DOI: https://doi.org/10.4213/mzm7707

Full text: PDF file (451 kB)
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English version:
Mathematical Notes, 2010, 87:4, 575–581

Bibliographic databases:

UDC: 517.5
Received: 09.02.2009

Citation: M. Sh. Shabozov, “Widths of Classes of Periodic Differentiable Functions in the Space $L_{2}[0,2\pi]$”, Mat. Zametki, 87:4 (2010), 616–623; Math. Notes, 87:4 (2010), 575–581

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. 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
    2. 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
    3. Shabozov M.Sh. Vakarchuk S.B., “On the best approximation of periodic functions by trigonometric polynomials and the exact values of the widths of function classes in $L_2$”, Anal. Math., 38:2 (2012), 147–159  crossref  mathscinet  zmath  isi  scopus
    4. Shabozov M.Sh., “Exact Jackson-Stechkin-type inequalities for $2\pi$-periodic functions in $L_2$ and widths of some classes of functions”, Ukr. Math. J., 63:10 (2012), 1633–1639  crossref  mathscinet  zmath  isi  scopus
    5. 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
    6. 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
    7. G. A. Yusupov, “Best polynomial approximations and widths of certain classes of functions in the space $L_2$”, Eurasian Math. J., 4:3 (2013), 120–126  mathnet
    8. 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
    9. 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
    10. 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
    11. 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
    12. 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
    13. Tukhliev K., “Neravenstva tipa Dzheksona-Stechkina dlya obobschennykh modulei nepreryvnosti i nekotorye ikh primeneniya”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:11 (2013), 861–868  elib
    14. Oliftaev N.F., “O znacheniyakh nekotorykh klassov periodicheskikh differentsiruemykh funktsii v prostranstve $L_2$”, Izvestiya Akademii nauk Respubliki Tadzhikistan. Otdelenie fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2013, no. 1(150), 21–31  elib
    15. 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
    16. 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
    17. S. B. Vakarchuk, M. Sh. Shabozov, M. R. Langarshoev, “On the best mean square approximations by entire functions of exponential type in $L_2(\mathbb R)$ and mean $\nu$-widths of some functional classes”, Russian Math. (Iz. VUZ), 58:7 (2014), 25–41  mathnet  crossref
    18. 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
    19. 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  elib  scopus
    20. 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
    21. 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
    22. Shabozov M.Sh. Yusupov G.A. Temurbekova S.D., J. Approx. Theory, 215 (2017), 145–162  crossref  mathscinet  zmath  isi  elib  scopus
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