Approximation by Stancu–Chlodowsky type λ-Bernstein operators

In this paper, we give some approximation properties by Stancu–Chlodowsky type λ-Bernstein operators in the polynomial weighted space and obtain the convergence properties of these operators by using Korovkin’s theorem. We also establish the direct result and the Voronovskaja type asymptotic formula.

The Dunkl generalization of Stancu type q-Szasz-Mirakjan-Kantorovich operators and some approximation results

In this paper, a Dunkl type generalization of Stancu type q-Szasz-Mirakjan-Kantorovich positive linear operators ´ of the exponential function is introduced. With the help of well-known Korovkin’s theorem, some approximation properties and also the rate of convergence for these operators in terms of the classical and second-order modulus of continuity, Peetre’s K-functional and Lipschitz functions are investigated.

Approximation by generalized Stancu type integral operators involving Sheffer polynomials

In this article, we give a generalization of integral operators which involves Sheffer polynomials introduced by Sucu and Buy¨ ukyazici. We obtain approximation properties of our operators with the help of the univer- ¨ sal Korovkin’s theorem and study convergence properties by using modulus of continuity, the second order modulus of smoothness and Peetre’s K-functional. We have also established Voronovskaja type asymptotic formula. Furthermore, we study the convergence of these operators in weighted spaces of functions on the positive semi-axis and estimate the approximation by using weighted modulus of continuity.

Approximation by q-analogue of modified Jakimovski-Leviatan-Stancu type operators

In this paper, we introduce the q-analogue of the Jakimovski-Leviatan type modified operators introduced by Atakut with the help of the q-Appell polynomials.We obtain some approximation results via the well-known Korovkin’s theorem for these operators.We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover,we study the rate of convergence in terms of modulus of continuity of these operators in aweighted space.

A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials

We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.

Hermite-Fejér type interpolation and Korovkin's theorem

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].