Statistical Convergence Estimates for (p, q)-Baskakov-Durrmeyer Type Operators

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Bézier variant of Bernstein–Durrmeyer blending-type operators

In this paper, we construct the Bézier variant of the Bernstein–Durrmeyer-type operators. First, we estimated the moments for these operators. In the next section, we found the rate of approximation of operators [Formula: see text] using the Lipschitz-type function and in terms of Ditzian–Totik modulus of continuity. The rate of convergence for functions having derivatives of bounded variation is discussed. Finally, the graphical representation of the theoretical results and the effectiveness of the defined operators are given.

On the q-Monotonicity Preservation of Durrmeyer-Type Operators

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or $$[0,\infty )$$ [ 0 , ∞ ) as the case may be. Recall that a 1-monotone function is nondecreasing, a 2-monotone one is convex, and for $$q>2$$ q > 2 , a q-monotone function possesses a convex $$(q-2)$$ ( q - 2 ) nd derivative in the interior of the interval. The operators are the Durrmeyer versions of Bernstein (including genuine Bernstein–Durrmeyer), Szász and Baskakov operators. As a byproduct we have a new type of characterization of continuous q-monotone functions by the behavior of the integrals of the function with respect to measures that are related to the fundamental polynomials of the operators.

Approximation of Functions by Dunkl-Type Generalization of Szász-Durrmeyer Operators Based on p , q -Integers

In this article, our main purpose is to define the p , q -variant of Szász-Durrmeyer type operators with the help of Dunkl generalization generated by an exponential function. We estimate moments and establish some direct results of the aforementioned operators. Moreover, we establish some approximation results in weighted spaces.

Exponential moments and simultaneous approximation properties for Durrmeyer type operators with weights of Szasz basis functions

The present paper deals with the approximation properties for exponential functions of general Durrmeyer type operators having the weights of Sz?sz basis functions. Here we give explicit expressions for exponential type moments by means of which we establish, for the derivatives of the operators, the Voronovskaja formulas for functions of exponential growth and the corresponding weighted quantitative estimates for the remainder in simultaneous approximation.

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>

On approximation of Bernstein-Durrmeyer-type operators in movable interval

In the present paper, we introduce a new type of Bernstein-Durrmeyer operators preserving linear functions in movable interval. The approximation rate of the new operators for continuous functions and Voronovskaja?s asymptotic estimate are obtained.