Zoltán Finta

On certain q-Durrmeyer type operators

Very recently Gupta and Heping [V. Gupta, W. Heping, The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math Meth. Appl. Sci. 31 (16) (2008) 1946–1955] introduced certain q-Durrmeyer type operators and established some approximation properties. In the present paper, we study some direct results for the certain q-Durrmeyer type operators.

Approximation properties of q-Baskakov operators

We establish direct estimates for the q-Baskakov operator introduced by Aral and Gupta in [2], using the second order Ditzian-Totik modulus of smoothness. Furthermore, we define and study the limit q-Baskakov operator.

A certain family of mixed summation-integral type operators

In the present paper, we study the mixed summation integral type operators having different weight functions, we obtain the rate of point wise convergence, an asymptotic formula of Voronovskaja type and some local direct results in terms of modulus of smoothness and modulus of continuity in ordinary and simultaneous approximation.

Direct and converse results for Stancu operator

In this paper we establish direct and inverse theorems for Stancu operator. Some other approximation properties of these operators are also given.

Bernstein-Schurer-Kantorovich operators based on q-integers

In this paper, we introduce a new Kantorovich type generalization of the q-Bernstein-Schurer operators defined in Muraru (2011). First, we give the basic convergence theorem and then obtain the local direct results for these operators, estimating the rate of convergence by using the modulus of smoothness and the Lipschitz type maximal function, respectively. We also obtain a Voronovskaja type theorem and investigate the statistical approximation properties of these operators with the help of a Korovkin type statistical approximation theorem given in Duman (2008).

Direct results for a certain family of summation–integral type operators

Abstract In the present paper, we establish some local and global direct results for a certain family of summation–integral operators B n . We also estimate the rate of convergence of B n ( f ) for a function f having derivatives of bounded variation.

Approximation properties of (p, q)-Bernstein type operators

Abstract We introduce a new generalization of the q-Bernstein operators involving (p, q)-integers, and we establish some direct approximation results. Further, we define the limit (p, q)-Bernstein operator, and we obtain its estimation for the rate of convergence. Finally, we introduce the (p, q)-Kantorovich type operators, and we give a quantitative estimation.

Bernstein type operators having 1 and x j as fixed points

For certain generalized Bernstein operators {Ln} we show that there exist no i, j ∈ {1, 2, 3,…}, i < j, such that the functions ei(x) = xi and ej (x) = xj are preserved by Ln for each n = 1, 2,… But there exist infinitely many ei such that e0(x) = 1 and ej (x) = xj are its fixed points.

Quantitative estimates for the Lupaş q-analogue of the Bernstein operator

Abstract We establish quantitative results for the approximation properties of the q-analogue of the Bernstein operator defined by Lupaş in 1987 and for the approximation properties of the limit Lupaş operator introduced by Ostrovska in 2006, via Ditzian-Totik modulus of smoothness. Our results are local and global approximation theorems.

Generalized Voronovskaja theorem for q-Bernstein polynomials

We prove a Floater type theorem for the q-Bernstein polynomials, establishing a generalized Voronovskaja theorem. Moreover, a quantitative variant of the obtained generalized Voronovskaja theorem for the q-Bernstein polynomials is also given. As a particular case, we recover the quantitative Voronovskaja theorem established by Videnskii (2005).