Local and global results for modified Szász–Mirakjan operators

In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.     

Certain positive linear operators with better approximation properties

The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.     

Asymptotic expansion of multivariate Kantorovich type operators

In this paper we study the asymptotic expansion of sequences of multivariate Kantorovich type operators and their partial derivatives. In particular, we obtain the complete expansion for the Kantorovich Bernstein operators on the simplex and for two Kantorovich type modifications of the Bleimann, Butzer and Hahn operators that we introduce in the paper. AMS subject classification 41A36     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.     

On Convergence Properties for a Class of Kantorovich Discrete Operators

Here we give some pointwise convergence theorems and asymptotic formulae of Voronovskaja type for a general class of Kantorovich discrete operators. Applications to the Kantorovich version of some discrete operators are given.     

Weighted statistical approximation by kantorovich type q-Szász-Mirakjan operators

In the present paper, we introduce a Kantorovich type modification of q-Szász-Mirakjan operators and obtain weighted statistical approximation properties of these operators. Also for introduced operators, we give a Voronovskaja type theorem related to q-derivatives.