The authors present a Bernstein-type operator L n which is given by a finite sum and defined on the space C (0,∞) It is shown that L n defines a positive linear operator such that L n f ( x ) tends pointwise on [0, ∞) to f ( x ) for n → ∞ Moreover, estimations for the rate of convergence of L n f ( x )− f ( x ) are established, measured in terms of the second modulus of continuity of f 

A Bernstein-type operator approximating continuous functions on the semi-axis

We give quantitative results for variation{diminishing splines, focusing on the case of equidistant knots. New direct inequalities are obtained, both in terms of the classical second modulus of continuity and in terms of the second Ditzian{Totik modulus. These new results are based upon a detailed analysis of the second moments and very recent theorems for positive linear operator approximation. The potential for simultaneous approximation is described by means of an estimate involving both the flrst and the second classical modulus of continuity. The topic of global smoothness preservation is also addressed. Furthermore, we discuss the degree of simultaneous approximation in the multivariate case, namely for Boolean sums and tensor products of Schoenberg splines.

On Variation{diminishing Schoenberg Operators: New Quantitative Statements ⁄

The purpose of this survey paper is to examine the applicabilities of the Banach-Steinhaus theorem in a version equipped with rates in the light of results obtained at Aachen during the past two years These lie in a variety of fields, namely in (a) Fourier analysis, (b) quadrature formulae, (c) mean ergodic theory, (d) algebraic approximation, (e) probability theory, (f) numerical analysis, and (g) signal theory In each case direct theorems involving rates of convergence are presented The hypotheses of the general theorem applied basically consist of two inequalities: one involves the rate of increase of the operator norm, the other a Jackson-type inequality

The Banach-Steinhaus Theorem with Rates, and Applications to Various Branches of Analysis

The main object of this paper is to construct new Durrmeyer type operators which have better features than the classical one. Some results concerning the rate of convergence and asymptotic formulas of the new operator are given. Finally, the theoretical results are analyzed by numerical examples.

Better Numerical Approximation by Durrmeyer Type Operators

Concerning the approximation of nonperiodic functions by positive linear methods the best-known operators are the Bernstein polynomials*) and the singular integral of Landau -Stieltjes (Landau polynomials). Another positive linear approximation process which is attracting increasing attention was introduced by W. MEYER -KONIG and K. ZELLER [2]/[3] in 1959/60; for functions f on [0,1) it was originally defined by 

$$ {M_n}\left( {f;x} \right): = {\left( {1 - x} \right)^n}\mathop \sum \limits_{k = 0}^\infty \;\;f\left( {\frac{k}{{k + n}}} \right)\left( {_k^{k + n - 1}} \right){\text{ }}{x^k},\quad n \in {\text{N}} $$

(1)

.

A Bibliography of the Bernstein Power Series Operators of Meyer — König and Zeller and Their Generalizations

This note is concerned with positive linear approximation processes φ n ( f ; t ) ( n = 1, 2, …) on C [ a , b ] which are exact for the functions l, t . A short proof will be given for the pointwise estimate of | f ( t ) − φ n ( f ; t )| ( t ∈ [ a , b ]) in terms of the second modulus of continuity, i.e. with increment depending on t .

H Esser

Papers

On pointwise convergence estimates for positive linear operators on C[a, b]