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Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szasz-Mirakyan Operator

Butzer P.L., Karsli H.

Commentationes Mathematicae

2009

Vol. 49, [Z] 1

33-58

This paper is devoted to a study of a Voronovskaya-type theorem for the derivative of the Bernstein-Chlodovsky polynomials and to a comparison of its approximation eectiveness with the corresponding theorem for the much better-known Szasz-Mirakyan operator. Since the Chlodovsky polynomials contain a factor bn tending to innity having a certain degree of freedom, these polynomials turn out to be generally more ecient in approximating the derivative of the associated function than does the Szasz operator. Moreover, whereas Chlodovsky polynomials apply to functions which are even of order O(exp(x^p)) for any p ≥ 1; the Szasz-Mirakyan operator does so only for p = 1; it diverges for p > 1. The proofs employ but rene practical methods used by Jerzy Albrycht and Jerzy Radecki ( in papers which are almost never cited ) as well as by further mathematicians from the great Poznań school.

Approximation by modified Favard operators

Rempulska L., Tomczak K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr 2

205-215

We introduce certain modification of the Favard and Favard-Durrmeyer operators and we give some approximation properties for them. The similar results for the modified Bernstein polynomials are given in [5].