SIMULTANEOUS APPROXIMATION BY BERNSTEIN-SIKKEMA OPERATORS

In this paper, we investigate the simultaneous approximation of Bernstein- Sikkema operators, and establish the direct and equivalent theorems by using the Ditzian-Totik modulus of smoothness.

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.

EQUIVALENT THEOREMS ON SIMULTANEOUS APPROXIMATION BY COMBINATIONSOF GAMMA OPERATORS

In this paper, some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωφλ^r（f,t）wφ^s（0≤λ≤1）are given. The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.

Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.

Weighted Approximation by Left Quasi-interpolants of Derivatives of Gamma Operators

In order to obtain much faster convergence, Miiller introduced the left Gamma quasi- interpolants and obtained an approximation equivalence theorem in terms of 2r wφ （f,t）p. Cuo extended the MiiUer＇s results to wφ^24 （f, t）∞. In this paper we improve the previous results and give a weighted approximation equivalence theorem.

Convergence of the q Analogue of Szász-Beta-Stancu Operators

In the present paper, we propose the q analogue of Szasz-Beta-Stancu operators. By estimate the moments, we establish direct results in terms of the modulus of smoothness. Investigate the rate of point-wise convergence and weighted approximation properties of the q operators. Voronovskaja type theorem is also obtained. Our results generalize and supplement some convergence results of the q-Szasz-Beta operators, thus they improve the existing results.

Bernstein-Sikkema算子及其导数的逼近性质

主要利用统一光滑模ω2φλ(f,t)讨论了Bernstein-Sikkema算子的一致逼近(λ=1)及点态逼近(0≤λ<1)问题,同时给出算子导数的一个估计.

On Approximation by Neural Networks with Optimized Activation Functions and Fixed Weights

Recently,Li[16]introduced three kinds of single-hidden layer feed-forward neural networks with optimized piecewise linear activation functions and fixed weights,and obtained the upper and lower bound estimations on the approximation accuracy of the FNNs,for continuous function defined on bounded intervals.In the present paper,we point out that there are some errors both in the definitions of the FNNs and in the proof of the upper estimations in[16].By using new methods,we also give right approximation rate estimations of the approximation by Li’s neural networks.

一类基于Pólya分布的修正Durrmeyer型算子的逼近

引进一类基于Pólya分布的修正Durrmeyer型算子,该算子具有常数保持与线性保持的性质。利用连续模、光滑模及K-泛函,讨论该算子的某些逼近性质,最后得出该算子的Voronovskaya型渐近公式。

Beta算子加权逼近的点态结果

引入一种改变的带权K-泛函,利用带权光滑模和带权主部光滑模的关系及带权光滑模与改变的带权K-泛函的等价性,讨论了Beta算子的点态带Jacobi权逼近正定理及等价定理.

Szasz-Durrmeyer算子逼近的强逆不等式

引入新的K-泛函K(f,t)β研究Szasz-Durrmeyer算子逼近的强逆不等式,从而得到了算子逼近的特征刻画.1)设f∈CB[0,∞),则存在常数R>1,当l≥Rn时,有K(f,1/n)β≤Cln.(‖Mnf-f‖β+‖Mlf-f‖β);2)设0<h<1/16,0<α≤2,0≤λ≤1,则对每个x∈[0,∞),存在n=n(x,h)及正整数m,满足△2hφλf(x)≤Chαnα/2(‖Mnf-f‖β+‖Mmnf-f‖β).

一类和积分型混合算子在L^p空间的逼近性质

研究了以Baskakov和Beta为基函数的一类和积分型混合算子,得到了在Lp(1≤p≤∞)空间逼近的正、逆定理以及等价定理.利用统一光滑模ωφ2λ(f,t)(0≤λ≤1),得到了点态逼近的等价定理.

Baskakov算子线性组合同时逼近的等价定理

本文建立了Baskakov算子线性组合同时逼近的等价定理

Bernstein算子的点态同时逼近

借助光滑模W2Ф（f，t）（Ф是一般步权函数），研究了Bernstein算子的点态同时逼近问题，给出了Bernstein算子同时逼近的等价定理，建立了其导数与光滑函数间的关系，对以前已有的结果予以补充和完善．

Baskakov算子的同时逼近

借助于Ditzian-Totik光滑模研究了Baskakov算子的同时逼近问题,给出了Baskakov算子同时逼近的正定理和等价定理.

一类新算子的同时逼近

引入一种新的正线性算子并研究它对于无界函数的同时逼近.设f∈Cβ[0,∞),r∈N,f(x)在[0,∞)存在r阶导数,则limn∞M(r)n,α(f(t),x)=f(r)(x);若f(r)(x)∈C(a-η,b+η)(η>0),则M(r)n,α(f,x)f(r)(x)在x∈[a,b]一致成立.设f∈Cβ[0,∞),f(x)在[0,∞)上存在r+2阶导数,则limn∞n[M(r)n,α(f,x)-f(r)(x)]=α[r(r+1)f(r)(x)+(2(r+1)x+r)f(r+1)(x)+x(1+x)f(r+2)(x)];若f(r+2)(x)∈Ca-η,b+η)(η>0),则上式在[a,b]一致成立.

Bernstein型算子线性组合的同时逼近等价定理

本文建立了 Bernstein算子及 Bernstein- Kantorovich算子线性组合的同时逼近的正、逆定理 。

关于Szsz-Durrmeyer-Bzier算子的点态逼近

Bzier型算子是一些著名算子的推广,已有研究成果主要是对有界变差函数的逼近,而对于应用光滑模研究其中心逼近定理的结果很少。本文利用一阶Ditzian-Totik模得到了Szsz-Durrmeyer-Bzier算子点态逼近的正、逆定理及等价定理这一完美的逼近结果。

Bernstein算子线性组合同时逼近的点态估计

借助于光滑模ωrφλ(f ,t) (0≤λ≤ 1 )给出了

Bernstein算子线性组合同时逼近的正逆定理

借助于r阶光滑模ωr(f,t)(是一般的步权函数)给出了Bernstein算子线性组合同时逼近的正逆定理。