Ulrich Abel

Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation

We study the approximation properties of beta operators of second kind. We obtain the rate of convergence of these operators for absolutely continuous functions having a derivative equivalent to a function of bounded variation.

Over-iterates of Bernstein's Operators: A Short and Elementary Proof

5. CONCLUDING REMARKS. The motivating question of this note, as to whether there is a modification of Thomae’s function that is differentiable on the irrationals, arose while the first author was teaching a one-semester course in real analysis. This question’s connection to Diophantine approximation theory was not discovered until after the realization of Proposition 3.2. After the submission of the current manuscript, the authors were informed that a slightly less general version of Proposition 4.2 can be found in [9, p. 232].

The complete asymptotic expansionfor a general Durrmeyer variant of the Meyer-Konig and Zeller operators

In the present paper, we consider a general Durrmeyer type of the Meyer-Konig and Zeller operators and derive the complete asymptotic expansion for these operators. Previous Durrmeyer-type variants of the Meyer-Konig and Zeller operators are covered by the present definition.

Asymptotic Expansion for Szász‐Mirakyan Operators

We obtain an asymptotic expansion of the well‐known Szasz‐Mirakyan operatorsWe obtain an asymptotic expansion of the well‐known Szasz‐Mirakyan operators

On the rate of convergence of bezier variant of Szasz-Durrmeyer operators

In the present paper, we introduce Szász-Durrmeyer-Bézier operators Mn,a(f,x), which generalize the Szász-Durrmeyer operators. Here we obtain an estimate on the rate of convergence of Mn,a(f,x) for functions of bounded variation. Our result extends and improves that of Sahai and Prasad[9] and Gupta and Pant[3].

Complete monotonicity and zeros of sums of squared Baskakov functions

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

Asymptotic approximation of functions and their derivatives by Müller’s Gamma operators

We obtain the complete asymptotic expansion of the image functions of Müller’s Gamma operators and of their derivatives. All expansion coefficients are explicitly calculated. Moreover, we study linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations of which special cases were recently introduced and investigated by other authors.

Asymptotic Approximation with a Sequence of Positive Linear Operators

The rate of pointwise convergence for a sequence of positive linear operators Ln approximating continuous functions on a finite interval is considered. The complete asymptotic expansion for the operators Ln as n tends to infinity is presented. It turns out that the central factorial numbers of first and second kind play an important role in the asymptotic expansion. The present work is an extension to that reported by Ivan and Raşa.

Complete asymptotic expansion for multivariate Bernstein-Durrmeyer operators and quasi-interpolants

We establish complete asymptotic expansions in terms of certain differential operators for the Bernstein-Durrmeyer operators with Jacobi weights on the d-dimensional simplices, and for their so-called natural quasi-interpolants. Our method extensively uses spectral properties of the involved operators. In particular, we prove new identities for the eigenvalues of the operators. We also show that the obtained asymptotic expansions can be differentiated term-by-term.

A Generalization of the Leibniz Rule

Abstract We present a generalization of the Leibniz Rule. One of its consequences is the celebrated Abel identity.