A class of linear positive operators in weighted spaces

Abstract

ABSTRACT. In this paper, we introduce a class of linear positive operators based on q-integers. For these operators we give some convergence properties in weighted spaces of continuous functions and present an application to differential equation related to q-derivatives. Furthermore, we give a Stancu-type remainder.

Some preservation properties of MKZ-Stancu type operators

“…Very recently, Erençin,İnce and Olgun [5] proposed a modification of the operators given by (1.1) based on q-integers and obtained some convergence properties of these operators in weighted spaces of continuous functions on [0, ∞) with the help of a weighted Korovkin type theorem. Furthermore, they also gave an application to functional differential equations and Stancu type remainder.…”

“…We remark that since P n,0 (α, 0) = 1 and P n,k (α, 0) = 0 for k ≥ 1 we have S n,α (f ; 0) = f (0). By taking into consideration this fact, a simple computation shows that the operators S n,α for α > 0 can also be represented as 5) where the operators M * n are given by (1.1) and…”

On generalized Baskakov-Durrmeyer-Stancu type operators

“…In [3], Wa and Khatoon studied the rate of convergence of the generalized Baskakov operators in terms of the modulus of continuity, and obtained a Voronovskaja type theorem and a direct estimate of these operators in terms of the Ditzian-Totik modulus of smoothness, respectively. In [4], Erencin and Bascanbaz-Tunca studied the weighted approximation properties and estimated the order of approximation in terms of the usual modulus of continuity for the operators (1). In 2008, Wa and Khatoon [5] obtained the convergence and a Voronovskaja type theorem for rst derivatives of generalized Baskakov operators for functions of one and two variables in exponential and polynomial weighted spaces.…”

ON A SEQUENCE OF KANTOROVICH TYPE OPERATORS VIA RIEMANN TYPE q-INTEGRAL

“…Thereafter many authors studied q-generalization of classical linear positive operators (see, for instance [2,3,9,10,12,14,17,18,20,28,30]). We now mention some works related to generalization of MKZ operators based on q-integers.…”

“…In [14], for q ∈ (0, 1) and every n ∈ N we proposed the following q-generalization of the operators M n (f, b n ; x) given by (1.1),…”