Genuine Bernstein Durrmeyer Operators Research Articles

Perturbed Bernstein-type operators

The present paper deals with modifications of Bernstein, Kantorovich, Durrmeyer and genuine Bernstein–Durrmeyer operators. Some previous results are improved in this study. Direct estimates for these operators by means of the first and second modulus of continuity are given. Also the asymptotic formulas for the new operators are proved.

Approximation by certain linking operators

In this paper, we improve the order of approximation of certain operators linking Bernstein and genuine Bernstein–Durrmeyer operators. Firstly, we obtain some direct results in terms of modulus of continuity and Voronovskaja type asymptotic formula for these operators. Finally, we give some numerical examples regarding the obtained theoretical results for new constructed operators.

Estimates for the differences of positive linear operators and their derivatives

The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Oxur approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, and Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of the first modulus of continuity. In order to analyze the theoretical results in the last section, we consider some numerical examples.

Convergence Properties of Certain Positive Linear Operators

The present paper deals with the modified positive linear operators that present a better degree of approximation than the original ones. This new construction of operators depend on a certain function $$\varrho $$ defined on [0, 1]. Some approximation properties of these operators are given. Using the first order Ditzian–Totik modulus of smoothness, some Voronovskaja type theorems in quantitative mean are proved. The main results proved in this paper are applied for Bernstein operators, Lupas operators and genuine Bernstein–Durrmeyer operators. By numerical examples we show that depending on the choice of the function $$\varrho $$ , the modified operator presents a better order of approximation than the classical ones.

$$\left( p,q\right) $$ p , q -Genuine Bernstein Durrmeyer operators

In the present article, we deal with the extension of quantum calculus based on two parameters, namely (p, q)-calculus. In the recent years the applications of quantum calculus in the field of approximation theory was considered widely. Here we propose (p, q)-variant of genuine Bernstein Durrmeyer operators and establish some direct results.

Power series of the operators $$U_n^{\varrho }$$ U n ϱ

We study power series of members of a class of positive linear operators reproducing linear function constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach.

Approximation properties of Bernstein–Durrmeyer type operators

This paper deals with the approximation of continuous functions by sequences of some modified Bernstein–Durrmeyer type operators that reproduce certain test functions. The orders of approximation of the new versions turn to be at least as good as the one of the genuine Bernstein–Durrmeyer operators. Moreover, by extrapolating techniques recently applied to the classical Bernstein operators, we present a one-parameter family of modified sequences of operators that reproduce certain polynomials and possess that popular genuine sequence as a limit case. Comparisons and some illustrative graphics are also presented.

The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein–Durrmeyer operators

We study the eigenstructure of a one-parameter class of operators $${U_{n}^{\varrho}}$$ of Bernstein–Durrmeyer type that preserve linear functions and constitute a link between the so-called genuine Bernstein–Durrmeyer operators U n and the classical Bernstein operators B n . In particular, for $${\varrho\rightarrow\infty}$$ (respectively, $${\varrho=1}$$ ) we recapture results well-known in the literature, concerning the eigenstructure of B n (respectively, U n ). The last section is devoted to applications involving the iterates of $${U_{n}^{\varrho}}$$ .

The Genuine Bernstein–Durrmeyer Operators Revisited

We present some direct and inverse results for the approximation by the genuine Bernstein–Durrmeyer operators Un. We also consider iterates of Un and discuss some convergence results towards the corresponding semigroup. Using the eigenstructure of the operators Un we give new proofs of some known qualitative results and obtain new quantitative estimates concerning the convergence of iterates towards the semigroup.

The Genuine Bernstein–Durrmeyer Operators and Quasi-Interpolants

In this paper we consider the so-called genuine Bernstein–Durrmeyer operators and define corresponding quasi-interpolants of order \({r \in \mathbb{N}_0}\) in terms of certain differential operators. These quasi-interpolants preserve all polynomials of degree at most r + 1. We analyse the eigenstructure of the differential operators and the quasi-interpolants and prove as main results an error estimate of Jackson–Favard type for sufficiently smooth functions and an upper bound for the error of approximation in the sup-norm in terms of an appropriate K-functional.

Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.

Estimates for Iterates of Positive Linear Operators Preserving Linear Functions

We consider iterates of certain (general) positive linear operators preserving linear functions and derive quantitative upper estimates in terms of weighted and non-weighted moduli of smoothness and related K-functionals. We show that corresponding lower estimates in terms of the classical moduli are not possible, while for the Ditzian–Totik modulus the situation can be different. The results can be applied to several well-known operators; we present here the Bernstein and the genuine Bernstein–Durrmeyer operators.

The Limiting Semigroup of the Genuine Bernstein–Durrmeyer Iterates

We provide a quantitative estimate for the degree of approximation of iterated genuine Bernstein–Durrmeyer operators to the members (T(t))t≥0 of their limiting semigroup.

Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: [formula omitted]-estimates

In the present note a general inequality for the degree of approximation of semigroups by iterates of commuting bounded linear operators on Banach spaces is given. Combining this with a recent quantitative Voronovskaja-type result applications to Durrmeyer operators with ultraspherical weights are derived. Our considerations include the genuine Bernstein–Durrmeyer operators.

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators Un by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by Un. Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of Un, compute the moments of T(t) and establish asymptotic formulas.