Abstract
In extremal problems of the theory of approximation of functions an important role is played be exact inequalities of the value of the best polynomial approximation by means of averaged values of the modules of continuity of higher orders of the derived functions. In this paper we present an inequality of type Ligun-two-sided estimate for the best weighted approximate analytic functions in the unit disc from the Bergman space 𝐵_2,𝛾 . The resulting inequalities allow us to establish new connections between the constructive and structural properties of the functions and for the corresponding classes of functions give an estimate from the top of the widths. The exact values of Bernstein, Kolmogorov, Gelfand, linear and projection n-widths of classes of analytic functions in unit discs defined by modules of continuity of higher orders of the derived functions in the space 𝐵_2,𝛾 averaged with positive weight are calculated
Highlights
In this paper we present an inequality of type Ligun-two-sided estimate for the best weighted approximate analytic functions in the unit disc from the Bergman space B2,γ
Наилучшее приближение аналитических функций в пространстве Бергмана // Дисс
Summary
Где m, n ∈ N, r ∈ Z+, 0 < p < ∞, φ(t) ≥ 0, 0 < t ≤ h, 0 < h ≤ π/n. Пусть X банахово пространство, S – единичный шар в X, N – выпуклое центральносимметричное подмножество X, Λn ⊂ X – n-мерное подпространство, Λn ⊂ X – подпространство коразмерности n, L : X – непрерывный линейный оператор, L⊥ : X → Λn – непрерывный оператор линейного проектирования. }︁ X, называют соответственно бернштейновским, гельфандовским, колмогоровским, линейным и проекционным n-поперечниками в пространстве X. Поскольку X – гильбертово пространство, то между перечисленными выше n-поперечниками выполняются соотношения [9]: bn(N, X) ≤ dn(N, X) ≤ dn(N, X) = λn(N, X) = Πn(N, X)
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