Methods of Approximation Theory

Approximation of (ψ, β)-differentiable functions by Weierstrass integrals

“…Furthermore, if β = r, r ∈ N, then f ψ β is the rth-order derivative of the function f, and the classes C ψ β,∞ are the well-known Sobolev classes W r . Following Stepanets [1], we denote the set of convex-downward sequences ψ(k) for which lim k→∞ ψ(k) = 0 by M. Without loss of generality, we assume that sequences ψ(k) from the set M are the restrictions of certain positive, continuous, convex-downward functions ψ(t) of continuous argument t ≥ 1 that tend to zero at infinity to the set of natural numbers. The set of these functions is also denoted by M. Thus, M = ψ(t) : ψ(t) > 0, ψ(t 1 ) − 2ψ t 1 + t 2 2 + ψ(t 2 ) ≥ 0 ∀t 1 , t 2 ∈ [1, ∞), lim t→∞ ψ(t) = 0 .…”

“…Further, we introduce the subset M 0 of the set M by using the following characteristic: Let ψ ∈ M and let η(t) = η(ψ; t) be the function related to ψ by the equality η(t) = η (ψ; t) = ψ −1 1 2…”

“…Below, we present several definitions and auxiliary statements due to Bausov [5] and Stepanets [1], which are used in what follows. [5].…”

Kantrovich Type Generalization of Meyer-Konig and Zeller Operators via Generating Functions

“…p 108 of [16]), we now define the following W ω class. We say that a function f ∈ C [0, A] belongs to…”

Approximation of functions defined on the real axis by operators generated by λ-methods of summation of their Fourier integrals

“…where U f x σ ( , , ) Λ are the operators defined by (2). First, we present several auxiliary definitions and statements necessary for what follows.…”