On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

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,     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

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.     

Pointwise weighted approximation by Bernstein operators

We consider the pointwise weighted approximation by Bernstein operators. The related weight functions are Jacobi weights . We obtain approximation equivalence theorems, using a weighted modulus of smoothness , which is an extension of~[5].     

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.     

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     

On simultaneous approximation of the Bernstein Durrmeyer operators

The present paper deals with the study of the rate of convergence of the Bézier variant of certain Bernstein Durrmeyer type operators in simultaneous approximation.     

On weighted approximation by rational operators for functions with singularities

We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [?1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia?[5].     

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.     

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     

Weighted simultaneous approximation with Baskakov type operators

We obtain inequalities for the weighted approximation error of Baskakov type operators and their derivatives. Such inequalities are valid for functions of polynomial growth and are expressed in terms of weighted moduli of continuity.     

Bernstein算子线性组合同时逼近的点态估计

借助于光滑模ωψ^rλ（f,t）（0≤λ≤1）给出了Bernstein算子线性组合同时逼近的点态结果。     

Nonlinear weighted best simultaneous approximation in Banach spaces

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction.     

Some approximation results for Durrmeyer operators

In the present paper, we obtain a sequence of positive linear operators which has a better rate of convergence than the Szász-Mirakian Durrmeyer and Baskakov Durrmeyer operators and their Voronovskaya type results.     

Bernstein-Kantorovich 算子线性组合同时逼近的正逆定理

借助光滑模ωtφ三(f,t)给出了Bernstein-Kantorovich 算子线性组合同时逼近的正逆定理,其中φ是一般步权函数,对已有的结果进行了补充和完善.