Heiner Gonska

Generalized Alomari Functionals

A generalized form of certain integral inequalities investigated by Guessab, Schmeisser and Alomari is given. The trapezoidal, midpoint, Simpson, Newton–Simpson rules are obtained as special cases. Also, inequalities for the generalized Alomari functional in terms of n-th order moduli are given and applied to some known quadrature rules.

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.

A new estimate for Hölder approximation by Bernstein operators

Abstract In this work we discuss the rate of simultaneous approximation of Holder continuous functions by Bernstein operators, measured by Holder norms with different exponents. We extend the known results on this topic.

Composite Bernstein cubature

We consider a sequence of composite bivariate Bernstein opera- tors and the cubature formula associated with them. The upper-bounds for the remainder term of cubature formula are described in terms of moduli of continu- ity of order two. Also we include some results showing how non-multiplicative the integration functional is.