Approximation Properties of a Certain Modification of Durrmeyer Operators

We study approximation properties of a new operator DM,1 n (f, x) introduced by Acu et al. in [Results Math 74:90, (2019)] for Lebesgue integrable functions in [0,1]. An error estimate by the Bezier variant of the operators DM,1 n (f, x)is also obtained for the functions of bounded variation. By relevant numerical examples, the orders of approximation by the operator DM,1 n (f, x) and the modified-Bernstein-Durrmeyer operator are also compared.

Better approximation of functions by genuine Bernstein-Durrmeyer type operators

The main object of this paper is to construct a new genuine Bernstein-Durrmeyer type operators which have better features than the classical one. Some direct estimates for the modified genuine Bernstein-Durrmeyer operator by means of the first and second modulus of continuity are given. An asymptotic formula for the new operator is proved. Finally, some numerical examples with illustrative graphics have been added to validate the theoretical results and also compare the rate of convergence.

APPROXIMATION THEOREMS FOR LIMIT $(p,q)-$BERNSTEIN-DURRMEYER OPERATOR

In the present paper, using the method developed in \cite{Finta1}, we prove the existence of the limit operator of the slight modification of the sequence of $(p,q)$-Bernstein-Durrmeyer operators introduced recently in \cite{Gupta1}. We also establish the rate of convergence of this limit operator.

Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators

For the qualitative results of uniform and pointwise approximation obtained in [8], we present here general quantitative estimates in terms of the modulus of continuity and of a K-functional, in approximation by the generalized multivariate Bernstein-Durrmeyer operator Mn,Γn,x, written in terms of Choquet integrals with respect to a family of monotone and submodular set functions, Γn,x, on the standard d-dimensional simplex. If d = 1 and the Choquet integrals are taken with respect to some concrete possibility measures, the estimate in terms of the modulus of continuity is detailed. Examples improving the estimates given by the classical operators also are presented.