Construction of a new family of Bernstein-Kantorovich operators

/pdf/semi-groups-of-operators-and-approximation-ebik2obms7.pdf

Semi-Groups of Operators and Approximation

In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators for bivariate functions using a new (p, q)-integral. We first estimate the moments and central moments. We give the uniform convergence of new operators, rate of convergence in terms of modulus of continuity. The approximations behaviours of the operators for functions having continuous partial derivatives and for functions belong to Lipschitz class are investigated as well.

Approximation by Bivariate ( p , q )-Bernstein–Kantorovich Operators

The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $
 -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $
 -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.

/pdf/on-kantorovich-modification-of-p-q-baskakov-operators-4wy1aj60up.pdf

On Kantorovich modification of (p,q)-Baskakov operators

Some approximation theorems VIA Statistical convergence

In this paper, we introduce new modifications of Szasz–Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p = 1, the previous results for q-Sz asz–Mirakyan operators are captured. Copyright © 2015 John Wiley & Sons, Ltd.

(p,q)-Generalization of Szász–Mirakyan operators

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Gruss type approximation theorem and the Gruss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.

Tuncer Acar

Papers

(p,q)-Generalization of Szász–Mirakyan operators

Construction of a new family of Bernstein-Kantorovich operators

Approximation by Bivariate ( p , q )-Bernstein–Kantorovich Operators

On Kantorovich modification of (p,q)-Baskakov operators

The new forms of Voronovskaya’s theorem in weighted spaces