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

Abstract

We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ 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 the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$.

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

“…The quantity e ⊥ m (F) X is called the best orthogonal trigonometric approximation of the class F in the space X. The quantities e ⊥ m (F) X for different functional classes F in the Lebesque spaces L q (T d ) as well as in some of their subspaces were investigated in many papers (see, e.g., [1,8,15,16,18,21,25,27,30]), where one can find a more detailed bibliography. Now we formulate two further needed statements.…”

“…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, ∞}.…”

The Problem of Best One Sided Approximation in Weighted

“…where is chosen so that The following property of Jackson kernels is well known (see e.g. [12, p. 193]) for (11) For put (12) Consider the function (13) and is a polynomial which understand the order of singledirectional approximation of and then, where (14) For the validity of ( 14) it is enough to choose in (13) any but for our next purposes we need Using we construct our one sided operators as follows:…”

“…[2020] Hybrid Block Successive Approximation for One-Sided Non-Convex Min-Max Problems Algorithms and Applications [12] and in [2021] Al-Jawari et al studied best one-sided multiplier approximation of unbounded functions by algebraic Polynomials operators in space ( ) by terms averaged modulus [13]. Furthermore, Fedunyk and Hembars'ka [14]2022 found best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables.…”