The best approximations and widths of the classes of periodical functions of one and several variables in the space $B_{\infty, 1}$

Abstract

We obtained exact-order estimates of the best approximations of classes B Ω ∞, θ of periodic functions of many variables and for classes B ω p,θ , 1 ≤ p < ∞ of functions of one variable by trigonometric polynomials with corresponding spectra of harmonics in the metric of space B ∞,1. We also found exact orders for the Kolmogorov, linear and trigonometric widths of the same classes in space B ∞,1 .

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

“…Similarly as in the proof of Theorem 5, the upper bound is obtained by approximating the functions f ∈ B Ω ∞,θ by trigonometric polynomials of the form (11) under the condition M ≍ 2 n n d−1 . The corresponding estimate is obtained in [15].…”

“…It is important to note that the estimate (19) is obtained by approximation using the linear method. More specifically, such a method in [15] used a sequence of linear operators {V Q n } ∞ n=1 , which to the function f ∈ B Ω ∞,θ put in correspondence the polynomial of the form…”

“…It is easy to prove that this function with the corresponding constant C 5 > 0 belongs to the class B Ω ∞,θ , 1 ≤ θ ≤ ∞. Next we need the estimation obtained in [15], namely…”

“…In this paper, we continue to study the approximative 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 [7,14,15,27,28]. As noted in these papers, the motivation to study the approximation characteristics (best approximation, widths, best M-term approximation etc.)…”

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

“…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.)…”