Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

Better degree of approximation by modified Bernstein-Durrmeyer type operators

<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id="M1">\begin{document}$ \tau(x), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is infinitely differentiable function on <inline-formula><tex-math id="M3">\begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau^{\prime }(x)&gt;0, \;\forall\;\; x\in[0, 1]. $\end{document}</tex-math></inline-formula> We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function <inline-formula><tex-math id="M5">\begin{document}$ \tau(x) $\end{document}</tex-math></inline-formula> leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [<xref ref-type="bibr" rid="b11">11</xref>].</p>

Better numerical approximation by λ-Durrmeyer-Bernstein type operators

The main object of this paper is to construct a new Durrmeyer variant of the ?-Bernstein type operators which have better features than the classical one. Some results concerning the rate of convergence in terms of the first and second moduli of continuity and asymptotic formulas of these operators are given. Moreover, we define a bivariate case of these operators and investigate the approximation degree by means of the total and partial modulus of continuity and the Peetre?s K-functional. A Voronovskaja type asymptotic and Gr?ss-Voronovskaja theorem for the bivariate operators is also proven. Further, we introduce the associated GBS (Generalized Boolean Sum) operators and determine the order of convergence with the aid of the mixed modulus of smoothness for the B?gel continuous and B?gel differentiable functions. Finally the theoretical results are analyzed by numerical examples.

A new modification of Durrmeyer type mixed hybrid operators

In 2008 V. Mihes¸an constructed a general class of linear positive operators generalizing the Szasz operators. In ´ this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.

Durrmeyer variant of q-Favard-Szász operators based on Appell polynomials

Karaisa [Karaisa, A., Approximation by Durrmeyer type Jakimoski Leviatan operators, Math. Method. Appl. Sci., DOI: 10.1002/mma.3650 (2015)] introduced the Durrmeyer type variant of Jakimovski-Leviatan operators based on Appell polynomials and studied some approximation properties. The aim of the present paper is to define the q analogue of these operators and establish the rate of convergence for a Lipschitz type space and a Lipschitz type maximal function for the Durrmeyer type variant of these operators. Also, we study the degree of approximation of these operators in a weighted space of polynomial growth and by means of weighted modulus of continuity