Widths of functional classes defined by majorants of generalized moduli of smoothness in the spaces ${\mathcal S}^p$

Abstract

Exact Jackson-type inequalities are obtained in terms of best approximations and averaged values of generalized moduli of smoothness in the spaces S p . The values of Kolmogorov, Bernstein, linear, and projective widths in the spaces S p are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

“…In the spaces L 2 , for classical moduli of smoothness inequality (20) was proved by Chernykh [15]. The inequalities of this type were also investigated in [31,32,37,38,25,35,16,6,36,5,3], etc.…”

“…functions, similar spaces were studied in the papers of Stepants and his followers, and they were denoted by S p [27,28,38,25,35], [29,Ch. 11], [3,26], etc. In [28], direct and inverse theorems for the approximation of functions from the spaces S p were proved in terms of their best approximations by trigonometric polynomials and moduli of smoothness of arbitrary positive orders.…”

“…In [28], direct and inverse theorems for the approximation of functions from the spaces S p were proved in terms of their best approximations by trigonometric polynomials and moduli of smoothness of arbitrary positive orders. In [3], exact Jackson-type inequalities in the spaces S p were obtained in terms of the best approximations of functions and the averaged values of their generalized moduli of smoothness as well as the exact values were found for widths of classes of 2π-periodic functions defined by certain conditions on the averaged values of their generalized moduli of smoothness.…”

Jackson-type inequalities and widths of functional classes in the Musielak-Orlicz type spaces

“…He found the exact values of the widths of such classes in the spaces L 2 in the case when the majorants Ω of the averaged values of the moduli of smoothness satisfied some constraints. The problem of finding the exact values of the widths in different spaces of functional classes of this kind was also studied in [3,15,23,25,33,35,37], etc.…”

“…Proof. The proof of Theorems 3.2 and 3.3 basically repeats the proof of corresponding theorems in the spaces S p (see [3,23]) and is adapted in accordance with the properties of the spaces S M . Based on Theorem 3.1, taking into account the definition of the set Ψ, for an arbitrary function f ∈ L ψ (ϕ, τ, µ, n) * M , we have…”