The generalization of Meyer-König and Zeller operators by generating functions

Abstract

In this paper, theorems are proved concerned with some approximation properties of generating functions type Meyer-König and Zeller operators with the help of a Korovkin type theorem. Secondly, we compute the rates of convergence of these operators by means of the modulus of continuity, Peetre's K-functional and the elements of the modified Lipschitz class. Also we introduce the rth order generalization of these operators and we obtain approximation properties of them. In the last part, we give some applications to the differential equations.  2005 Elsevier Inc. All rights reserved.

Kantrovich Type Generalization of Meyer-Konig and Zeller Operators via Generating Functions

“…Now, we give the following Korovkin type theorem for the test functions proved by Altın, Dogru and Taşdelen [1].…”

“…The Meyer-König and Zeller operators were also generalized in [4] by Dogru. A Stancu type generalization of the operators (1) have been studied by Agratini [2]. Dogru andÖzalp [5] studied a Kantorovich type generalization of the operators.…”

“…Using statistically convergence, a Kantorovich type generalization of Agratini's operators have been studied by Dogru, Duman and Orhan in [6]. Furthermore Altın, Dogru and Taşdelen [1] introduced a generalization of the operators (1) via linear generating functions as follows:…”

“…The Authors studied approximation properties of the operators (2) under the following conditions (see [1]):…”

Approximation of functions of two variables by certain linear positive operators

“…Moreover we define an rth order generalization of L α i ,β j m,n , j = 1, 2, on R A extending the results of Kirov [14] and KirovPopova [15] to the linear positive operators L α i ,β j m,n , j = 1, 2, of functions of two variables and study its approximation properties. The rth order generalization of some kind of linear positive operators may also be found in [1,9].…”