TheL p-saturation results weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators

The aim of the present paper is to prove new equivalence results and L_p-stauration results on weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators (including results in [7]). One of the main tools and crucial estimates managing the converse results is given by a direct modified Vornorskaja theorem which uses the third order weighted modulus of smoothness. Supported by NSF of Hebei Province.     

<Emphasis Type="Italic">L</Emphasis><Superscript>p</Superscript> approximation by general Bernstein-Durrmeyer operator defined on simplex

As a generalization of the Bernstein-Durrmeyer operators defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decomposition way. From the theorems the characterization of L^p approximation behavior is derived     

Approximation Theorems of Bernstein-Durrmeyer Operators in L_M~（Ba） Spaces

With the weighted modulus of smoothness as a metric,we prove the direct and the inverse theorems of approximation by Bernstein-Durrmeyer operators in LBa M spaces. Especially an approximation equivalent theorem of the operators is also obtained.     

Lp APPROXIMATION BY GENERAL BERNSTEIN-DURRMEYER OPERATOR DEFINED ON SIMPLEX

As a generalization of the Bernstein-Durrmeyer operatora defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decom-position way. From the theorems the characterization of Lp approximation behavior is derived.     

Ba空间中的多元加权光滑模与Bernstein Durrmeyer算子

该文引进Ba空间多元加权光滑模,推广L^p空间的DitzianTotik模, 证明该模与K泛函的等价性. 作为应用,讨论定义在单纯形上多元Bernstein-Durrmeyer算子与多元加权光滑模之间的关系. 即以多元加权光滑模为尺度, 建立Bernstein-Durrmeyer算子在Ba空间逼近阶的上界与下界估计.     

Bernstein-Durrmeyer算子拟中插式的逼近

最近,为了得到更快的逼近速度,人们引入了某些著名算子的拟中插式.我们研究了Bernstein-Durrmeyer算子的拟中插式Mn(2r-1)(f,x),用Ditzian-Totik模得到了它们的正、逆定理和等价定理.这里.     

An application of Bernstein-Durrmeyer operators

In the present paper, we find that the Bemstein-Durrmeyer operators, besides their better applications in approximation theory and some other fields, are good tools in constructing translation network. With the help of the de la Vallee properties of the Bernstein-Durrmeyer operators a sequence of translation network operators is constructed and its degree of approximation is dealt.     

ASYMPTOTIC APPROXIMATION BYBERNSTEIN-DURRMEYER OPERATORS AND THEIR DERIVATIVES

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

Asymptotic approximation by Bernstein-Durrmeyer operators and their derivatives

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

Orlicz空间内正系数代数多项式倒数对非负连续函数的逼近

本文研究了Bernstein-Durrmeyer代数多项式倒数对非负连续函数在Orlicz空间中的逼近问题.利用光滑模和K-泛函等工具,获得了收敛速度的估计,所得的结果比Lp空间内的相应结果具有拓展的意义.     

加权范数下Bernstein－Durrmeyer多项式的一个饱和结果

研究Ｂｅｒｎｓｔｅｉｎ－Ｄｕｒｒｍｅｙｅｒ多项式的加权逼近并建立其饱和定理．     

The lower estimate for linear positive operators,I

A method to prove lower estimates for linear operators is introduced. As a result the best lower estimate for certain convolution operators, for the multivariate Bernstein-Durrmeyer operators in part I and the Bernstein polynomial operators in part II (see [10]), are obtained.Communicated by Hubert Berens     

Approximation by a class of modified Szász-Durrmeyer operators

We modify Szász-Durrmeyer operators by means of three-diagonal generalized matrix which overcomes a difficulty in extending a Berens-Lorentz result to the Szász-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators inL _p are also presented by Ditzian-Totik modulus of smoothness.This project is supported by Zhejiang Provincial Foundation of China.     

Inverse theorems for some multidimensional operators

In this paper we introduce a new K-functional which is especially useful for non-Feller operators. Inverse theorems for multidimensional Bernstein-Sikkema and Bernstein-Durrmeyer operators are given on a simplex and on a cube. Supported by National Science Foundation of China.     

Steckin-Marchard-Type Inequalities for Some Durrmeyer Operators

In this paper, we will derive some Steckin-Marchard-type inequalities for Bernstein-Durrmeyer and Szasz-Durrmeyer operators.     

A NOTE ON BERNSTEIN TYPE OPERATORS

In this note we give a counterexample to a result of Z.Ditzian and K.Ivanov.A directtheorm on C[0,1]is also presented for Kantorovich operators and Bernstein-Durrmeyer op-erators.     

Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions

We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S.N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.     

Steckin-Marchard-Type Inequalities for Some Durrmeyer Operators

In this paper, we will derive some Steckin-Marchard-type inequalities for Bernstein-Durrmeyer and Szasz-Durrmeyer operators.     

关于Bernstein-Durrmeyer算子的Stechkin-Marchaud型不等式

主要目的是利用K-泛函来研究Bernstein-durrmeyer型算子的Stechkin-Marchaud型不等式，由此不等式，我们推广子B-D算子关于ω^2ψλ（f,t)的逆结果。