Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type
In this paper, it is considered the extremal problem of finding the exact constants in inequalities of Jackson – Stechkin type between the best approximations of periodic differentiable functions f ∈ L (r) 2 [0, 2π] by trigonometric polynomials, and the average values with a positive weight ϕ moduli...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2013-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/177 |
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| author | G. A. Yusupov |
| author_facet | G. A. Yusupov |
| author_sort | G. A. Yusupov |
| collection | DOAJ |
| description | In this paper, it is considered the extremal problem of finding the exact constants in inequalities of Jackson – Stechkin type between the best approximations of periodic differentiable functions f ∈ L (r) 2 [0, 2π] by trigonometric polynomials, and the average values with a positive weight ϕ moduli of continuity of mth order ωm(f (r) , t), belonging to the space Lp, 0 < p ≤ 2. In particular, the problem of minimizing the constants in these inequalities over all subspaces of dimension n, raised by N.P. Korneychuk, is solved. For some classes of functions defined by the specified moduli of continuity, the exact values of n-widths of class L (r) 2 (m, p, h; ϕ) := f ∈ L (r) 2 : Z h 0 ω p m(f (r) ;t)2 ϕ(t)dt 1/p Z h 0 ϕ(t)dt −1/p ≤ 1 are found in the Hilbert space L2, and the extreme subspace is identified. In this article, the results are shown which are the extension and the generalization of some earlier results obtained in this line of investigation. |
| format | Article |
| id | doaj-art-995548343c06448e8bd8ef3e6540a2be |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2013-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-995548343c06448e8bd8ef3e6540a2be2025-08-20T04:00:26ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172013-10-0120510611610.18255/1818-1015-2013-5-106-116171Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin TypeG. A. Yusupov0Tajik National UniversityIn this paper, it is considered the extremal problem of finding the exact constants in inequalities of Jackson – Stechkin type between the best approximations of periodic differentiable functions f ∈ L (r) 2 [0, 2π] by trigonometric polynomials, and the average values with a positive weight ϕ moduli of continuity of mth order ωm(f (r) , t), belonging to the space Lp, 0 < p ≤ 2. In particular, the problem of minimizing the constants in these inequalities over all subspaces of dimension n, raised by N.P. Korneychuk, is solved. For some classes of functions defined by the specified moduli of continuity, the exact values of n-widths of class L (r) 2 (m, p, h; ϕ) := f ∈ L (r) 2 : Z h 0 ω p m(f (r) ;t)2 ϕ(t)dt 1/p Z h 0 ϕ(t)dt −1/p ≤ 1 are found in the Hilbert space L2, and the extreme subspace is identified. In this article, the results are shown which are the extension and the generalization of some earlier results obtained in this line of investigation.https://www.mais-journal.ru/jour/article/view/177best approximationsmodule of continuity of mth ordern-widths |
| spellingShingle | G. A. Yusupov Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type Моделирование и анализ информационных систем best approximations module of continuity of mth order n-widths |
| title | Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type |
| title_full | Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type |
| title_fullStr | Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type |
| title_full_unstemmed | Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type |
| title_short | Exact Values of Widths of Some Functional Classes in L2 and Minimization of the Constants in Inequalities of Jackson – Stechkin Type |
| title_sort | exact values of widths of some functional classes in l2 and minimization of the constants in inequalities of jackson stechkin type |
| topic | best approximations module of continuity of mth order n-widths |
| url | https://www.mais-journal.ru/jour/article/view/177 |
| work_keys_str_mv | AT gayusupov exactvaluesofwidthsofsomefunctionalclassesinl2andminimizationoftheconstantsininequalitiesofjacksonstechkintype |