On Weighted Approximation by Modified Bernstein Operators for Functions with Singularities

Della Vecchia et al. （see [2]） introduced a kind of modified Bernstein operators which can be used to approximate functions with singularities at endpoints on [0,1]. In the present paper, we obtain a kind of pointwise Stechkin-type inequalities for weighted approximation by the modified Bemsetin operators.

Pointwise Approximation for the Iterated Boolean Sums of Bernstein Operators

In this paper, with the help of modulus of smoothness ω2r(f,t), we discuss the pointwise approximation properties for the iterated Boolean sums of Bernstein operator B n and obtain direct and inverse theorems when 1-1/r ≤λ≤ 1, r ∈N.

REFERENCE FUNCTIONAL AND CHARACTERISTIC SPACE FOR LAGRANGE AND BERNSTEIN OPERATORS

This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.

A WAY OF CONSTRUCTING TRANSLATION NETWORK OPERATORS BY BERNSTEIN OPERATORS

In the present paper, we provide a way of constructing translation network operators by BernsteinDurrmeyer operators.

Approximation of Generalized Bernstein Operators

This paper is devoted to study direct and converse approximation theorems of the generalized Bemstein operators Cn （f, sn,x） via so-called unified modulus ωφλ^2 （f,t）, 0 ≤ λ ≤1. We obtain main results as followsωφλ^2 （f,t）=O（t^α）←→｜Cn（f,sn,x）-f（x）｜=O（（n^-1/2δn^1-λ（x））^α）,where δn^2（x）=max{φ^2（x）,1/n} and 0〈α〈2.

The Simultaneous Approximation Average Errors for Bernstein Operators on the r-Fold Integrated Wiener Space

For weighted approximation in Lp-norm,we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the r-fold integrated Wiener space.

On the rate of convergence of two generalized Bernstein type operators

In this paper,we introduce the Bézier variant of two new families of generalized Bernstein type operators.We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity.By means of construction of suitable functions and the method of Bojanic and Cheng,we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

A NEW PROPERTY OF BERNSTEIN-KANTOROVIC OPERATORS

In the present paper, we prove that the Bernstein-Kantorovic operators have the ability of preserving translation property in both C and Lp norms.

A NOTE ON BERNSTEIN TYPE OPERATORS

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

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.

BERNSTEIN'S FIRST SUMMABLE OPERATORS ON THE SPECIAL REAL ORTHOGONAL GROUP SO(n)

We study the structure of Bernstein's first summable operators and show that they con- verge uniformly to continuons functions on the special real orthogonal group SO(n)in this paper.In addition,we have also discussed the approximation degree to a class of function Lipα(0<α≤1)on SO(n).

一类Bernstein型算子在C_Q空间上的逼近性质

本文研究了由Stancu提出的Bernstein型逼近算子P_(n、s)在C_Ω空间的逼近性质,建立了等价逼近定理,从而统一地处理加权逼近和依范逼近中的等价性结论.

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 w^rφλ （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.

On Approximation of Nonbounded Continuous Functions

This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are analysed and studied. This paper gives a certain theorem as a general rule to approximate any nonbounded continuous functions.

Weighted approximation by Bernstein quasiinterpolants for functions with singularities

We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities.