Approximative characteristics of the Nikol'skii-Besov-type classes of periodic functions in the space $B_{\infty,1}$

Abstract

We obtained the exact order estimates of the orthowidths and similar to them approximative characteristics of the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ of periodic functions of one and several variables in the space $B_{\infty,1}$. We observe, that in the multivariate case $(d\geq2)$ the orders of orthowidths of the considered functional classes are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for orthowidths of the corresponding functional classes there are the ordinary partial sums of their Fourier series. Besides, we note that in the univariate case the estimates of the considered approximative characteristics do not depend on the parameter $\theta$. In addition, it is established that the norms of linear operators that realize the order of the best approximation of the classes $B^{\Omega}_{p,\theta}$ in the space $B_{\infty,1}$ in the multivariate case are unbounded.

Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables

“…Proof. First, we establish the upper bound in (13). We show that this estimate is realized by approximation of functions f ∈ B Ω 1,θ by their step hyperbolic Fourier sums…”

“…In this paper, we continue to study the approximation characteristics of the classes B Ω p,θ of periodic functions of several variables and classes B ω p,θ of one variable in the space B ∞,1 , which norm is stronger than the norm in L ∞ . We recall that some approximative characteristics of functional classes in the space B ∞,1 were studied in [8,13,16,17,[31][32][33]41]. As noted in these papers, the motivation to study the approximation characteristics (best approximation, widths, best M-term approximation, etc.)…”

Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions

“…In the number of papers [2,7,8,11,[24][25][26][27]31], the questions concerning approximation of classes of periodic multivariate functions with mixed smoothness (the classes of Nikol'skii-Besov-type B r p,θ , Sobolev classes W r p,α and some their analogs) in the normed spaces with slightly modified norms comparing to the norm of B q,1 , q ∈ {1, ∞}, were investigated. As a result of the investigations, it was revealed that in many situations the obtained estimates of considered approximative characteristics differ in order from the estimates of corresponding characteristics in the spaces L q , q ∈ {1, ∞}.…”