On a Problem of Gonska

This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.     

Approximation by a class of modified Bernstein-Durrmeyer operators

We modify Bernstein-Durrmeyer operators by means of digonal matrix which overcome a difficulty in extending a Berens-Lorentz result to the Bernstein-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators in L_p are also presented by Ditzian-Totik modulus of smoothness. Supported by Zhejiang Provincial Foundation of China     

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.     

APPROXIMATION BY A CLASS OF MODIFIED SZASZ-DURRMEYER OPERATORS

ThisprojectissupportedbyZhejiangProvincialFoundationofChina.1.IntroductionForjEC[0,1]ther-thBernsteinpolynomialisdefinedbyItwasshownbyH.BerensandG.G.Lorentz([2]in1972)thatif0相似文献   

Shepard-Lagrange算子的逼近

本文建立了Shepard-Lagrange算子逼近的正逆定理,证明了可以利用高阶光滑模来刻画Shepard-Lagrange算子的逼近性质.从而说明了Shepard-Lagrange算子比一般的Shepard算子具有更好的逼近性质.进一步,所用光滑模的阶梯函数非常广泛,这是多项式逼近所不具有的.     

Global smoothness preservation with second order modulus of smoothness

We establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator.     

Derivatives of multidimensional Bernstein operators and smoothness

We characterize the directional derivatives of multidimensional Bernstein operators by a new measure of smoothness. This task is carried out by means of establishing the relation between the asymptotic behavior of the derivatives and the smoothness of the functions they approximate. The obtained results generalize the corresponding ones for univariate Bernstein operators.     

多元Bernstein算子的导数与函数的光滑性

利用一个新的光滑性度量刻画多元Bernstein算子方向导数的特征,建立Bernstein算子的导数与逼近函数光滑性之间的等价关系。同时,一个关于一元Bernstein算子的相应结果被推广到多元情形。     

Equivalence theorem on weighted simultaneous Lp-approximation by means of Szasz-Kantorovich operators and Baskakov-Kantorovich operators

The intention of this paper is to give direct and converse results on weighted simultaneous approximation by means of Szász-Kantorovich operators and Baskakov-Kantorovich operators in L_p-norms using the weighted Ditzian-Totik modulus of smoothness. Supported by the foundation of Zhejiang province     

Smoothness Properties of Modified Bernstein-Kantorovich Operators

In this article, we consider modified Bernstein-Kantorovich operators and investigate their approximation properties. We show that the order of approximation to a function by these operators is at least as good as that of ones classically used. We obtain a simultaneous approximation result for our operators. Also, we prove two direct approximation results via the usual second-order modulus of smoothness and the second-order Ditzian-Totik modulus of smoothness, respectively. Finally, some graphical illustrations are provided.     

Construction of Real-Valued Wavelets by Symmetry

We characterize the higher orders of smoothness of functions in C[0, 1] by Bernstein polynomials and Kantorovich operators. This task is carried out by means of the rate of convergence for combinations of these operators and the behavior of their derivatives.     

On Global Smoothness Preservation by Bernstein Operators

We investigate the global smoothness preservation by Bernstein operators measured by second-order modulus of smoothness and give a partial answer for a conjecture raised by H. H. Gonska in 1998.     

Equivalent theorems on simultaneous approximation by combinations of Gamma operators

In this paper,some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωr(4)λ(f,t)w(4)s(O≤λ≤1)are given.The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.     

Bernstein型算子线性组合加Jacobi权逼近及高阶导数的等价定理

本文利用加权Ditzian-Totik光滑模证明Bernstein型算子的线性组合加权逼近阶估计和等价定理;同时,研究加Jacobi权下Benstein型算子的高阶导数与所逼近函数光滑性之间的关系.     

On an approximation family of discrete polynomial operators

We present a new family of linear discrete polynomial operators giving a Timan type approximation theorem for functions of arbitrary smoothness. Using this we construct two families of operators of this kind to extend Freud type approximation results to functions of higher smoothness.     

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.     

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.     

Some approximation properties of a Durrmeyer variant of q‐Bernstein–Schurer operators

In this paper, we will propose a Durrmeyer variant of q‐Bernstein–Schurer operators. A Bohman–Korovkin‐type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.     

Equivalent theorems on simultaneous approximation by combinations of Bernstein operatores

In this paper we give equivalent theorems on simultaneous approximation for the combinations of Bernstein operators by r-th Ditzian-Totik modulus of smoothness ωτψλ (f, t)(0 ≤λ≤ 1). We also investigate the relation between the derivatives of the combinations of Bernstein operators and the smoothness of derivatives of functions.     

Bernstein 算子的同时逼近

借助于D itzian-T otik光滑模研究了Bernstein算子的同时逼近问题,给出了Bernstein算子同时逼近的正定理和等价定理.