Ioan Gavrea

Asymptotic Behaviour of the Iterates of Positive Linear Operators

We present a general result concerning the limit of the iterates of positive linear operators acting on continuous functions defined on a compact set. As applications, we deduce the asymptotic behaviour of the iterates of almost all classic and new positive linear operators.

The Approximation of The Continuous Functions by Means of Some Linear Positive Operators

AbstractIn this paper we give positive answers to some problems posed by H.Gonska, respectively A.Lupaş. We show here that there exist positive linear operators Hn : C[0, 1] → πn satisfying

Full length article: On the iterates of positive linear operators

We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed.

On a conjecture concerning the sum of the squared Bernstein polynomials

We obtain a new representation of the sum of the squared Bernstein polynomials and use it to validate a conjecture asserting that this sum is a convex function. The result is extended to some other classical approximation operators.

Global smoothness preservation and the variation-diminishing property

In the center of our paper are two counterexamples showing the independence of the concepts of global smoothness preservation and variation diminution for sequences of approximation operators. Under certain additional assumptions it is shown that the variation-diminishing property is the stronger one. It is also demonstrated, however, that there are positive linear operators giving an optimal pointwise degree of approximation, and which preserve global smoothness, monotonicity and convexity, but are not variationdiminishing.

The iterates of positive linear operators preserving constants

Abstract In this note we introduce a simple and efficient technique for studying the asymptotic behavior of the iterates of a large class of positive linear operators preserving constant functions.

The Bernstein Voronovskaja-type theorem for positive linear approximation operators

We prove that the classical Bernstein Voronovskaja-type theorem remains valid in general for all sequences of positive linear approximation operators.

Variation on Butzer's problem: Characterization of the solutions

Abstract The so-called Butzer problem had been a challenge to the approximation community for decades until its first constructive solution was given by Cao and Gonska at the end of the 1980s. Butzers original problem can be turned into a “strong form”, namely a pointwise one. In the present note, we characterize, among other things, and in an easy way, certain operators which solve Butzers problem in this strong and most demanding version.

Positive Linear Operators with Equidistant Nodes

Abstract In the present paper, the approximation power of positive linear operators with equidistant nodes is investigated. New pointwise estimates are given in terms of first and second order moduli of continuity, showing that for positive operators having uniform Jackson orders of approximation one may expect interpolation at the endopoints. In particular, the first solution to Butzers problem with equidistant nodes is given. A negative result yields an explanation why Bernstein operators and some of their modifications are optimal in a certain sense.

An extremal property for a class of positive linear operators

We generalize a recent result of de la Cal and Carcamo concerning an extremal property of Bernstein operators.