Bernstein-Stancu算子的加權(quán)逼近

錢(qián) 程， 滕旦霞， 虞旦盛

(1.杭州師范大學(xué)理學(xué)院，浙江 杭州 310036; 2.杭州汽車(chē)高級(jí)技工學(xué)校基礎(chǔ)部，浙江 杭州 310012)

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Bernstein-Stancu算子的加權(quán)逼近

錢(qián) 程1， 滕旦霞2， 虞旦盛1

(1.杭州師范大學(xué)理學(xué)院，浙江 杭州 310036; 2.杭州汽車(chē)高級(jí)技工學(xué)校基礎(chǔ)部，浙江 杭州 310012)

文章考慮了Bernstein-Stancu算子對(duì)具有奇性函數(shù)加Jacobi權(quán)的加權(quán)逼近問(wèn)題，建立了加權(quán)逼近的正、逆定理.

Bernstein-Stancu 型算子；加權(quán)逼近；奇性函數(shù)

0 引言和預(yù)備知識(shí)

前述關(guān)于Bernstein算子加Jacobi權(quán)的加權(quán)逼近結(jié)果都是針對(duì)連續(xù)函數(shù)而言的,而且權(quán)函數(shù)中的參數(shù)一般有上、下界限制(要求0

定理1 設(shè)a,b>0,0≤λ≤1.則對(duì)f∈Cw,有

值得注意的是,上述結(jié)論中權(quán)函數(shù)的參數(shù)范圍是a,b>0,沒(méi)有上界限制.為了提高逼近階,Yu構(gòu)造出了能夠逼近Cw中函數(shù)的Bernstein算子的線性組合[8]和Bernstein擬中插式[9]. Yu等[10]還利用截?cái)嗟姆椒?gòu)造出能夠逼近具有內(nèi)部奇性函數(shù)的修正Bernstein算子.

需要指出的是,Vechhia等[6]所構(gòu)造的算子并非正線性算子,對(duì)于保持目標(biāo)函數(shù)的幾何性質(zhì)等也有不足之處.為此,本文考慮Bernstein算子的一種重要的推廣形式—Bernstein-Stancu型算子的加權(quán)逼近性質(zhì).Bernstein-Stancu多項(xiàng)式算子是Stancu[11]首次引入的:

其中α,β為非負(fù)常數(shù),且有0≤α≤β.顯然當(dāng)α=β=0時(shí),該式即為通常的Bernstein多項(xiàng)式.Bernstein-Stancu算子是一種正線性算子,已經(jīng)被廣泛應(yīng)用于數(shù)學(xué)和計(jì)算機(jī)科學(xué)領(lǐng)域.

本文將證明在α，β>0時(shí)Bernstein-Stancu算子可以較好地逼近Cw中的函數(shù).以下總是假設(shè)α,β>0.事實(shí)上，本文有如下結(jié)論：

定理2 設(shè)f(x)∈Cw,a,b>0,0≤λ≤1,則對(duì)n=2,3,…,存在正常數(shù)C使得

定理3 設(shè)f(x)∈Cw,a,b>0,則對(duì)任何0<θ<1,有

1 引理及其證明

證明 記

先證明

(1)

其中兩個(gè)量p～q指的是存在兩個(gè)正常數(shù)C1和C2使得C2q≤p≤C1q.分幾種情況證明式(1).

且有

從而

當(dāng)k=n時(shí),類(lèi)似可證得

這樣,

綜上所述，可知式(1)成立.

另一方面,對(duì)任意γ,δ>0,成立[12]:

(2)

利用式(1)和(2),得

引理2 對(duì)于任意γ≥0,有

(3)

證明 由文[13]知

(4)

利用式(4)及

即得

(5)

由式(2)知

(6)

類(lèi)似地,有

(7)

利用式(5)，(6)，(7)知引理3此時(shí)成立.

因此，引理3此時(shí)也成立.

證明 注意到

因此,

利用式(1)的證明方法,可以證明當(dāng)n>2時(shí),有

這樣，利用式(2),就有

2 結(jié)論及其證明

定理2的證明 定義

Kφλ(f，t)w～wφλ(f，t)w,0≤λ≤1.

(8)

利用式(8)，對(duì)任意固定的n，x和λ，存在g(x)∈W(λ),使得

(9)

(10)

(11)

因此，只需要證明以下不等式：

(12)

利用式(3)，(10)和(11),得

綜合以上討論可知

定理3的證明 充分性由定理2即知.必要性由引理3，引理4，知名的Berens-Lorentz引理和文[1]中的方法易得.

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Weighted Approximation of Bernstein-Stancu Operators

QIAN Cheng1, TENG Danxia2, YU Dansheng1

(1. School of Science, Hangzhou Normal University, Hangzhou 310036, China; 2. Department of Public Education,Hangzhou Automobile Advanced Technician’s School, Hangzhou 310012, China)

The paper considerd the weighted approximation problem of Berstein-Stancu operators with singular function and Jacobi weight, and established the direct and converse theorems about weighted approximation.

Bernstein-Stancu type operators; weighted approximation; singular function

2016-05-04

虞旦盛(1976—),男,教授,主要從事函數(shù)逼近論研究.E-mial:dsyu_math@163.com

10.3969/j.issn.1674-232X.2016.06.014

O174.41 MSC2010: 41A25; 41A35

A

1674-232X(2016)06-0636-07