Abstract:
Assume that A(U) is the set of functions analytic in the disk U:={z:|z|<1}, L(r)2:=L(r)2(U) for r∈N is the class of functions f∈A(U) such that f(r)∈L(r)2, and W(r)L2 is the class of functions f∈L(r)2 satisfying the constraint ‖f(r)‖≤1. We find exact values for mean-square approximations of functions f∈W(r)L2 and their successive derivatives f(s) (1≤s≤r−1, r≥2) in the metric of the space L2. A similar problem is solved for the class W(r)2(Km,Ψ) (r∈Z+, m∈N) of functions f∈L(r)2 such that the K-functional of their rth derivative satisfies the condition Km(f(r),tm)≤Ψ(tm),0<t<1, where Ψ is some increasing majorant and Ψ(0)=0.
Citation:
M. Sh. Shabozov, M. S. Saidusajnov, “Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 258–272
\Bibitem{ShaSai19}
\by M.~Sh.~Shabozov, M.~S.~Saidusajnov
\paper Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 2
\pages 258--272
\mathnet{http://mi.mathnet.ru/timm1640}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-2-258-272}
\elib{https://elibrary.ru/item.asp?id=38071620}
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https://www.mathnet.ru/eng/timm/v25/i2/p258
This publication is cited in the following 15 articles:
M. Sh. Shabozov, A. A. Shabozova, “O sovmestnom priblizhenii nekotorykh klassov funktsii v prostranstve Bergmana B2”, Izv. vuzov. Matem., 2024, no. 6, 80–88
M. Sh. Shabozov, A. A. Shabozova, E. U. Kadamshoev, “Value of n-width of some classes of analytic functions in the Bergman space B2”, Moscow University Mathematics Bulletin, 79:3 (2024), 112–121
M. Sh. Shabozov, A. A. Shabozova, “On Simultaneous Approximation of Certain Classes of Functions in the Bergman Space B2”, Russ Math., 68:6 (2024), 68
M. Sh. Shabozov, D. K. Tukhliev, “On mean–square approximation of functions in Bergman space B2 and value of widths of some classes of functions”, Ufa Math. J., 16:2 (2024), 66–75
M. Sh. Shabozov, R. A. Karimzoda, “K-funktsionaly i tochnye znacheniya n-poperechnikov
nekotorykh klassov funktsii v prostranstve Xardi”, Tr. IMM UrO RAN, 30, no. 4, 2024, 301–308
Mirgand Sh. Shabozov, Muqim S. Saidusajnov, “On widths of some classes of analytic functions in a circle”, Ural Math. J., 10:2 (2024), 121–130
Kh. M. Khuromonov, “O nailuchshem sovmestnom priblizhenii funktsii v prostranstve Bergmana B2”, Izv. vuzov. Matem., 2023, no. 5, 71–81
M. Sh. Shabozov, “On the Best Simultaneous Approximation in the Bergman Space B2”, Math. Notes, 114:3 (2023), 377–386
Muqim S. Saidusajnov, “Some inequalities between the best simultaneous approximation and the modulus of continuity in a weighted Bergman space”, Ural Math. J., 9:2 (2023), 165–174
Kh. M. Khuromonov, “On the Best Simultaneous Approximation of Functions in the Bergman Space B2”, Russ Math., 67:5 (2023), 50
D. K. Tukhliev, “Neravenstvo tipa Kolmogorova v prostranstve Bergmana B2 i nekotorye ego prilozheniya”, Chebyshevskii sb., 24:5 (2023), 228–236
Kh. M. Khuromonov, M. Sh. Shabozov, “Neravenstva tipa Dzheksona — Stechkina mezhdu nailuchshimi sovmestnymi polinomialnymi priblizheniyami i odnoi kharakteristikoi gladkosti v prostranstve Bergmana”, Vladikavk. matem. zhurn., 24:1 (2022), 109–120
M. Sh. Shabozov, Z. Sh. Malakbozov, “On the best polynomial approximation in Hardy space”, Russian Math. (Iz. VUZ), 66:11 (2022), 97–109
M. Sh. Shabozov, M. S. Saidusainov, “Srednekvadraticheskoe priblizhenie nekotorykh klassov funktsii kompleksnogo peremennogo ryadami Fure v vesovom prostranstve Bergmana B2,γ”, Chebyshevskii sb., 23:1 (2022), 167–182
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