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This article is cited in 23 scientific papers (total in 23 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
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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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This publication is cited in the following articles:
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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
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Shabozov M.Sh. Yusupov G.A., “Widths of certain classes of periodic functions in $L_2$”, J. Approx. Theory, 164:7 (2012), 869–878
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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
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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
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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
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M. Sh. Shabozov, K. Tukhliev, “Best Polynomial Approximations and the Widths of Function Classes in $L_{2}$”, Math. Notes, 94:6 (2013), 930–937
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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
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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
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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
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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
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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
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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
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Tukhliev K., “Neravenstva tipa Dzheksona-Stechkina dlya obobschennykh modulei nepreryvnosti i nekotorye ikh primeneniya”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:11 (2013), 861–868
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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
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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
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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
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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
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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
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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
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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
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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
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Shabozov M.Sh. Yusupov G.A. Temurbekova S.D., J. Approx. Theory, 215 (2017), 145–162
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Vakarchuk S.B., “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
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