Asymptotic Properties of Bernstein–Durrmeyer Operators

It is known that Szász–Durrmeyer operator is the limit, in an appropriate sense, of Bernstein–Durrmeyer operators. In this paper, we adopt a new technique that comes from the representation of operator semigroups to study the approximation issue as mentioned above. We provide some new results on approximating Szász–Durrmeyer operator by Bernstein–Durrmeyer operators. Our results improve the corresponding results of Adell and De La Cal (Comput Math Appl 30:1–14, 1995).     

Local Approximation by Generalized Baskakov–Durrmeyer Operators

The paper deals with general Baskakov–Durrmeyer operators containing several previous definitions as special cases. The main results include the local rate of convergence, which is proved based on a representation of the kernel functions in terms of Jacobi polynomials and the complete asymptotic expansion for the sequence of these operators. In obtaining the expansion for simultaneous approximation, a key step is the use of a combinatorical identity for derivatives with weights.     

Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ ^d , we proved that for f∈L ^p (W _κ ;T ^d ) with 1<p<∞,

The Genuine Bernstein–Durrmeyer Operators Revisited

We present some direct and inverse results for the approximation by the genuine Bernstein–Durrmeyer operators U _n . We also consider iterates of U _n and discuss some convergence results towards the corresponding semigroup. Using the eigenstructure of the operators U _n 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.     

Approximation by Szasz-Mirakjan Type Operators in L''''

M.M.Derriennic discussed the properties of Bernstein-Durrmeyer operators,M. Heilmann solved the saturation situation and the author obtained the characte-rization of their order of approximation.As extending Kantorovich polynomials inL_p[0,1]to Szász-Mirakjan-Kantorovich operators in L_p[0,∞)by V.Totik,We in-troduce a new class of Szász-Mirakjan type operators:     

Approximation by Jakimovski–Leviatan Type Operators on a Complex Domain

The present paper deals with the study of approximation by complex Stancu type generalization of Jakimovski–Leviatan type operators on a parabolic domain subset of complex plane by using the methods of Dressel et al. (Pacific J Math 13(4):1171–1180, 1963).     

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 Complete Asymptotic Expansion for Bernstein–Durrmeyer Operators with Jacobi Weights

The purpose of this paper is the investigation of the local asymptotic behavior of the Bernstein-Durrmeyer polynomials and their derivatives with respect to Jacobi-weights. The main result is the complete asymptotic expansion for these polynomials and their derivatives. All coefficients are calculated explicitely.     

Approximation by Kantorovich-Szász Type Operators Based on Brenke Type Polynomials

In this article, we give a generalization of the Kantorovich-Szász type operators defined by means of the Brenke type polynomials introduced in the literature and obtain convergence properties of these operators by using Korovkin’s theorem. Some graphical examples using the Maple program for this approximation are given. We also establish the order of convergence by using modulus of smoothness and Peetre’s K-functional and give a Voronoskaja type theorem. In addition, we deal with the convergence of these operators in a weighted space.     

Approximation by Modified Durrmeyer-Bernstein Operators

In this paper, we study the problem of simultaneous approximation by these operators and give out exact estimate. The following main results are obtained. Theorem I If f∈C~r[0,1], then, for every x∈[0,1], we have     

Approximation by Bernstein–Durrmeyer-type operators in compact disks

In this paper, the order of simultaneous approximation and Voronovskaja-type theorems with quantitative estimate for complex Bernstein–Durrmeyer-type polynomials attached to analytic functions on compact disks are obtained. Our results show that extension of the complex Bernstein–Durrmeyer-type polynomials from real intervals to compact disks in the complex plane extends approximation properties.     

Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

We consider a combination of Chlodowsky polynomials with generalized Szasz operators involving Charlier polynomials. We give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, we construct a GBS (Generalized Boolean Sum) operator of bivariate Chlodowsky–Szasz–Charlier type and estimate the order of approximation in terms of mixed modulus of continuity.     

GBS Operators of Lupaş–Durrmeyer Type Based on Polya Distribution

In this paper, we study an extension of the bivariate Lupa?–Durrmeyer operators based on Polya distribution. For these operators we get a Voronovskaja type theorem and the order of approximation using Peetre’s K-functional. Then, we construct the Generalized Boolean Sum operators of Lupa?–Durrmeyer type and estimate the degree of approximation in terms of the mixed modulus of smoothness.     

On Approximation of Functions by Generalized Abel–Poisson Operators

We find the exact constant in the principal term of the deviation of a function in the Zygmund class from the generalized Abel-Poisson operators determined by the Fourier series over the trigonometric system with particular summation factors.     

Statistical Approximation by Double Poisson–Cauchy Singular Integral Operators

In this paper we show that it is possible to approximate a continuous and 2π-periodic function on the disk centered at origin with radius π by means of double Poisson–Cauchy singular integral operators which do not need to be positive in general. Our results cover not only the classical approximation but also the statistical approximation process.     

Asymptotic Property of Approximation to x~αsgnx by Newman Type Operators

The approximation of |x| by rational functions is a classical rational problem.This paper deals with the rational approximation of the function x~αsgnx, which equals |x| ifα=1.We construct a Newman type operator r_n(x) and show {|x~αsgnx-r_n(x)|}～Cn~(-(α/2)e-(2nα)~(1/2)), where C is a constant depending onα.