Characterization Theorems for the Approximation by a Family of Operators

Abstract

The intention of this paper is to study a family of positive linear approximation operators relating to most of the well known Bernstein-type operators. These operators depend on a parameter. We give some characterization theorems to show that the operators corresponding to different parameters can be quite different. The direct and converse results make use of the Ditzian Totik modulus of smoothness.1996 Academic Press, Inc.

Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions

“…The U ̺ n , 0 < ̺ < ∞, turn out to be the weak limits of certain operators studied earlier by Mache and Zhou [11]. As in their article, Euler's Beta function plays a fundamental role in our construction.…”

“…Next we show that the operators U ̺ n are the limit case of the operators studied by Mache and Zhou [11], which we denote by P ̺,a,b n . These operators are given, for…”

Local and Global Approximation Theorems for Positive Linear Operators

“…In order to investigate an inverse to (13), a lemma will be needed first. Lemma 1 generalizes a lemma of Becker [1, p. 138] for finite intervals.…”

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

“…In addition 19) and therefore the operator B defined in (2.17) and its domain D(B) may be rewritten in an apparently different way as…”

“…Rather recently an interesting generalization of the operators M n in terms of the so-called Jacobi weights has been developed in [7,8] (see also [18,19]). …”