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This article is cited in 25 scientific papers (total in 25 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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Linking options:
http://mi.mathnet.ru/eng/mz8821https://doi.org/10.4213/mzm8821 http://mi.mathnet.ru/eng/mz/v90/i5/p764
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This publication is cited in the following articles:
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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
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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
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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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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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Yusupov G.A., “O nailuchshikh srednekvadraticheskikh priblizheniyakh na vsei osi tselymi funktsiyami eksponentsialnogo tipa”, Doklady Akademii nauk Respubliki Tadzhikistan, 56:3 (2013), 192–195
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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., “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
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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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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
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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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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
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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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M. Sh. Shabozov, A. D. Farozova, “Tochnoe neravenstvo Dzheksona–Stechkina s neklassicheskim modulem nepreryvnosti”, Tr. IMM UrO RAN, 22, no. 4, 2016, 311–319
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K. Tukhliev, “Srednekvadraticheskoe priblizhenie funktsii ryadami Fure–Besselya i znacheniya poperechnikov nekotorykh funktsionalnykh klassov”, Chebyshevskii sb., 17:4 (2016), 141–156
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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
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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
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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
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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
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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
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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
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M. Sh. Shabozov, “Nekotorye voprosy approksimatsii periodicheskikh funktsii trigonometricheskimi polinomami v $L_2$”, Chebyshevskii sb., 20:4 (2019), 385–398
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M. Sh. Shabozov, E. U. Kadamshoev, “Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space”, Math. Notes, 110:2 (2021), 248–260
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