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 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 ${U_{n}^{\varrho}}$ .     

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).     

Approximation Properties by Bernstein–Durrmeyer Type Operators

In this paper, in order to make the convergence faster to a function being approximated, we modify the Bernstein–Durrmeyer type operators, which were introduced in Abel et al. (Nonlinear Anal Ser A Theory Methods Appl 68(11):3372–3381, 2008). The modified operators reproduce the constant and linear functions. The operators discussed here are different from the other modifications of Bernstein type operators. The Voronovskaja type asymptotic formula with quantitative estimate for a new type of complex Durrmeyer polynomials, attached to analytic functions in compact disks is obtained. Here, we put in evidence the overconvergence phenomenon for this kind of Durrmeyer polynomials, namely the extensions of approximation properties with exact quantitative estimates, from the real interval [0, 1/3] to compact disks in the complex plane.     

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.     

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.     

The Bernstein–Orlicz norm and deviation inequalities

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.     

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<∞,

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.     

多元Bernstein—Durrmeyer算子的正逆定理

本文给出单形Ｓ上多元Ｂｅｒｎｓｔｅｉｎ－Ｄｕｒｒｍｅｙｅｒ算子在连续函数空间Ｃ（Ｓ）的强型正定理的积分型估计式弱型逆定理从而建立了算子逼近特征刻划等价定理。     

修正Durrmeyer－Bernstein算子的L_p逼近

本文建立修正了Ｄｕｒｒｍｅｙｅｒ－Ｂｅｒｎｓｔｃｉｎ算子的逼近等价定理。     

二元Bernstein—Durrmeyer算子的若干性质

对于[0,1]上的实值可积函数 f,J.L.Durrmeyer 引进一种新型的 Bernstein 算子M_n(f,x)=(n 1)P_(nk)(x)∫_0~1P_(nk)(t)f(t)dt,其中 P_(nk)(x)=x~k(1-x)~(n-k),其中 P_(nk)(x)=x~k(1-x)~(n-k),这里 0≤x≤1,n=0,1,2,…在文[2]中,M.M.Derriennie 又进一步讨论了它的逼近性质.在本文中,我们把 M.M.Derriennie 的某些结果推广到多元的情形,得到了一系列结果.     

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.