In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {Bn,q(f)} for 0 0, α ≠ 1, and the orders do not depend on α, unlike the classical case.

The rate of convergence of q-bernstein polynomials for 0 strict inequality q strict inequality 1

In this paper, we introduce $(p,q)$
 -gamma operators which preserve $x^{2}$
 , we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems about Lipschitz functions are obtained. The estimates on the rate of convergence and some weighted approximation theorems of the operators are also obtained. Furthermore, the Voronovskaja-type asymptotic formula is also presented.

/pdf/p-q-p-q-gamma-operators-which-preserve-x-2-x-2-5baiq5p5hg.pdf

( p , q ) $(p,q)$ -gamma operators which preserve x 2 $x^{2}$

Our purpose is to introduce a two-parametric $(p, q)$
 -analogue of the Stancu-Beta operators. We study approximating properties of these operators using the Korovkin approximation theorem and also study a direct theorem. We also obtain the Voronovskaya-type estimate for these operators. Furthermore, we study the weighted approximation results and pointwise estimates for these operators.

/pdf/on-p-q-analogue-of-two-parametric-stancu-beta-operators-2kae3l1gt2.pdf

On ( p , q ) -analogue of two parametric Stancu-Beta operators

The main aim of this article is to introduce a new type of - Chlodowsky and - Szasz-Durrmeyer hybrid operators on weighted spaces. To this end, we give approximation properties of the modified new - Hybrid operators. Moreover, in the weighted spaces, we examine the rate of convergence of the modified new - Hybrid operators by means of moduli of continuity. In addition, we derive Voronovskaja’s type asymptotic formula for the related operators.

/pdf/the-q-chlodowsky-and-q-szasz-durrmeyer-hybrid-operators-on-49ccutgdsm.pdf

The q -Chlodowsky and q -Szasz-Durrmeyer Hybrid Operators on Weighted Spaces

In the present paper, we introduce q-Sz´asz Mirakyan operators by taking the weight function of q- Chlodowsky-Durrmeyer operators on C[0;1) and investigate their approximation properties. We give weighted  approximation theorem.

/pdf/approximation-by-composition-of-q-szasz-mirayan-and-q-4h9vgm0mdj.pdf

Approximation by Composition of q-SZASZ-Mirayan and q-Durrmeyer-Chlodowsky Operators

In the present paper, we introduce the q-Szasz-Durrmeyer operators and justify a local approximation result for continuous functions in terms of moduli of continuity. We also discuss a Voronovskaya type result for the q-Szasz-Durrmeyer operators.

On q-Szász-Durrmeyer operators

/pdf/on-certain-q-phillips-operators-lwh3gcuwep.pdf

On certain $q$-Phillips operators

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On q-Szász-Durrmeyer operators

On certain $q$-Phillips operators