Ali Aral

Applications of q-calculus in operator theory

Introduction of q-calculus.- q-Discrete operators and their results.- q-Integral operators.- q-Bernstein type integral operators.- q-Summation-integral operators.- Statistical convergence of q-operators.- q-Complex operators.

A generalization of Szász-Mirakyan operators based on q-integers

In this paper, we introduce a generalization of Szasz-Mirakyan operators based on q-integers, that we call q-Szasz-Mirakyan operators. Depending on the selection of q, these operators are more flexible than the classical Szasz-Mirakyan operators while retaining their approximation properties. For these operators, we give a Voronovskaya-type theorem related to q-derivatives. Furthermore, we obtain convergence properties for functions belonging to particular subspaces of C[0,~) and give some representation formulas of q-Szasz-Mirakyan operators and their rth q-derivatives.

Generalized q-Baskakov operators

In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.

Convergence of the q analogue of Szász-Beta operators

In the present paper we introduce the q analogue of the well known Szasz-Beta operators [11]. We also establish the approximation properties of these operators and estimate convergence results. In the end we propose an open problem.

Bleimann, Butzer, and Hahn operators based on the -integers.

We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes-integers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we define a generalization of these new operators and study the uniform convergence of them.

on q-Baskakov type operators

Abstrac t . In the present paper we introduce two g-analogous of the well known Baskakov operators. For the first operator we obtain convergence property on bounded interval. Then we give the montonity on the sequence of q-Baskakov operators for n when the function / is convex. For second operator, we obtain direct approximation property on unbounded interval and estimate the rate of convergence. One can say that , depending on the selection of q, these operators are more flexible then the classical Baskakov operators while retaining their approximation properties.

On Pointwise Convergence of q-Bernstein Operators and Their q-Derivatives

Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their q-derivatives. These theorems are given in terms of the modulus of continuity of q-derivative of f which is the main interest of this article. It is also shown that our results hold for continuous functions although those are given for two and three times continuously differentiable functions in classical case.

On the q analogue of Stancu-Beta operators

Abstract In the present work we introduce the q analogue of well known Stancu-Beta operators. We estimate moments and establish direct results in terms of the modulus of continuity. We also present an asymptotic formula.

On Kantorovich Process of a Sequence of the Generalized Linear Positive Operators

We define the Kantorovich variant of the generalized linear positive operators introduced by Ibragimov and Gadjiev in 1970. We investigate direct approximation result for these operators on p-weighted integrable function spaces and also estimate their rate of convergence for absolutely continuous functions having a derivative coinciding a.e., with a function of bounded variation.

Some approximation properties of q-Baskakov–Durrmeyer operators

Abstract Very recently Aral and Gupta [1] introduced q analogue of Baskakov–Durrmeyer operators. In the present paper we extend the studies, we establish the recurrence relations for the central moments and obtain an asymptotic formula. Also in the end we propose modified q -Baskakov–Durrmeyer operators, from which one can obtain better approximation results over compact interval.