Margareta Heilmann

Kantorovich Operators of Order k

This article is concerned with the k-th order Kantorovich modification of the classical Bernstein operators B n , namely, , where D k f is the derivative of order k and I k f is an antiderivative of order k of the function f. These operators are most useful in simultaneous approximation. We give detailed expressions of the moments up to order four of and estimates of ratios of moments. Next, they are used to refine Voronovskayas result for derivative approximation by B n . The properties of as approximation operators are also investigated.

Linear Combinations of Genuine Szász–Mirakjan–Durrmeyer Operators

We study approximation properties of linear combinations of the genuine Szasz–Mirakjan–Durrmeyer operators which are also known as Phillips operators. We prove a full quantitative Voronovskaja-type theorem generalizing and improving earlier results by Agrawal, Gupta, and May. A Voronovskaja-type result for simultaneous approximation is also established. Furthermore global direct theorems for the approximation and weighted simultaneous approximation in terms of the Ditzian–Totik modulus of smoothness are proved.

Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: C2-estimates

In the present note a general inequality for the degree of approximation of semigroups by iterates of commuting bounded linear operators on Banach spaces is given. Combining this with a recent quantitative Voronovskaja-type result applications to Durrmeyer operators with ultraspherical weights are derived. Our considerations include the genuine Bernstein-Durrmeyer operators.

kth Order Kantorovich Modification of Linking Baskakov-Type Operators

In 1957 Baskakov introduced a general method for the construction of positive linear operators depending on a real parameter c. The so-called genuine Baskakov–Durrmeyer-type operators form a class of operators reproducing the linear functions, interpolating at (finite) endpoints of the interval, and having other nice properties. In this paper we consider a nontrivial link between Baskakov-type operators and genuine Baskakov–Durrmeyer-type operators. We establish explicit representations for the images of monomials and for the moments; they are useful, e.g., in studying asymptotic formulas.

On the Decomposition of Bernstein Operators

Let F n be the linear operators on C[0, 1] defined by , where B n are the classical Bernstein operators and are Beta operators. This decomposition of B n was investigated in Gonska et al. [6]. Although the operators F n are not positive, they have quite interesting properties. We obtain new results concerning the convergence of the sequence (F n ). An associated quadrature formula with positive coefficients and equidistant knots is investigated, as well as the eigenstructure of F n .

Rodriguez-type Representation for the Eigenfunctions of Durrmeyer-type Operators

In this paper we consider a generalization of different variants of Durrmeyer- type modifications of Baskakov and Meyer- König and Zeller operators. We prove a general result concerning the commutativity of these operators with certain differential operators. Prom this result a Rodriguez- type formula for the eigenfunctions follows as a corollary.

On Simultaneous Approximation by Optimal Algebraic Polynomials

In this paper the order of bestapproximation by algebraic polynomials is related to the order of weighted simultaneous approximation. A special case of a Markov-Bernstein type inequality proved by Ditzian and Totik in a very general fashion is used.

Complements to Voronovskaya’s Formula

We generalize some known results concerning Voronovskaya-type formulas for the composition of two linear operators acting on an arbitrary Banach space.

A Nice Representation for a Link Between Bernstein-Durrmeyer and Kantorovich Operators

In this paper we consider a link between Bernstein-Durrmeyer operators and Kantorovich operators depending on a parameter \(\rho \). We prove a nice representation by using the classical Bernstein polynomials and generalize the results for k-th order Kantorovich modifications.