Bernstein-Sikkema算子逼近

第19卷　增刊数学研究与评论V o l.19Supp 1999年4月JOU RNAL O F M A TH E M A T I CAL R ESEA RCH AND EXPO S IT I ON A p r.1999Bern ste i n-Sikkema算子逼近Ξ熊庆良(绍兴鲁迅学院,浙江312000)　曹飞龙(宁夏大学数学系,银川750021)摘　要:研究Bernstein2Sikkem a算子的逼近问题,得到强型正定理和弱型逆定理,改进了文献[1]的结果.关键词:Bernstein2Sikkem a算子,光滑模,逼近.分类号:AM S(1991)41A10,41A36 CL C O174.41文献标识码:A 文章编号:10002341X(1999)增刊202612051　引　言记权函数Υ(x)=(x(1-x))12.对于f∈C[0,1],定义带权光滑模[2]ΞrΥ(f,t)=sup0<h≤t∃r hΥf与通常光滑模Ξr(f,t)=sup0<h≤t∃r h f ,其中f =m axx∈[0,1]f(x) ,∃r h f(x)=∑rk=0rk(-1)k f(x+(r2-k)h),x±r2h∈[0,1], 0 其它.引进K2泛函K rΥ(f,t r)=infg∈C r[0,1]{ f-g +t r Υr g(r) },K r(f,t r)=infg∈C r[0,1]{ f-g +t r g(r) },则有[2]M-1ΞrΥ(f,t)≤K rΥ(f,t r)≤MΞrΥ(f,t),(1.1)M-1Ξr(f,t)≤K r(f,t r)≤MΞr(f,t),(1.2)这里及以下的M均表与f,n无关的正常数,但在不同处其值一般不同.熟知,对于f∈C[0,1],其对应的B ern stein算子为Ξ收稿日期:1997201230B n (f ;x )=∑nk =0pn ,k(x )f (kn),　n ∈N ,其中p n ,k (x )=n kx k (1-x )n -k,　x ∈[0,1].P .C .Sikkem a [3]修改B ern stein 算子为如下的B ern stein 2Sikkem a 算子L n (f ;x )=∑nk =0p n ,k (x )f (k n +Α(n )),其中{Α(n )}是仅与n 有关的非负数列,且li mn →∞Α(n )n=0.显然,当Α(n )≡0时,B ern stein 2Sikkem a算子就是B ern stein 算子.对于B ern stein 2Sikkem a 算子,Sikkem a[3](亦可见[6])讨论了它在C[0,1]中的一些基本逼近性质.最近,李松[1]研究了该算子的逼近正逆定理,得到结果:定理A 设0≤Α(n )≤M ,f ∈C [0,1],n ≥4,则L n f -f ≤M n-1(∫1 2n-12Ξ2Υ(f ,t )t3d t +E 0(f )),其中E n (f )=inf { f -P n :P n 为阶数不超过n 的多项式的全体,n ∈N 0}.定理B 设0≤Α(n )≤M ,f ∈C [0,1],Β>0,则Ξ2Υ(f ,1n)≤M n-1(∑nk =1(nk )ΒL k f -f +n Βf ). 这里有必要指出:第一,本文将取消限制“0≤Α(n )≤M ”,而在条件Α(n )≥0,且li mn →∞Α(n )n=0之下研究B ern 2stein 2Sikkem a 算子逼近的正逆定理,这符合该算子的原始定义.第二,文献[1]的引言中写着:“由于当Α(n )>0时,B ern stein 2Sikkem a 算子不是保线性的,所以给出它的强正定理非常困难.”认为,B ern stein 2Sikkem a 算子(Α(n )>0)的非线性保持并不妨碍给出它的强型正定理.本文利用B ern stein 2Sikkem a 算子与B ern stein 算子的关系,就较简捷地得到了该算子逼近的强型正定理.第三,对定理B (逆定理)作了两点改进:一是去掉了和项“n Βf ”;二是抹去了和项“∑nk =1(n k )Β L k f -f ”中大于1的因子“(n k)Β”.这样就在较大程度上提高了定理B 的精度.本文的主要结果是定理1　设f ∈C [0,1],Α(n )≥0,且li mΑ(n )n=0,则L n f -f ≤M (Ξ2Υ(f ,1n)+Ξ(f ,Α(n )n)+1nf ),其中Ξ(f ,t )=Ξ1(f ,t )是f 的连续模.定理2　设f ∈C [0,1],Α(n )≥0,且li m n →∞Α(n )n=0,则Ξ2Υ(f ,1n)≤M n-1∑nk =1Lkf -f ,Ξ(f ,1n)≤M n-1(∑nk =1Lkf -f + f ). 推论1　若Α(n )≡0,0<Β<1,且f ∈C [0,1],则 L n f -f =O (n -Β)ΖΞ2Υ(f ,t )=O (t 2Β).推论2　若Α(n )≠o (1),0<Β<1,且f ∈C [0,1],则Ξ2Υ(f ,t )=O (t 2Β)Ξ(f ,t )=O (tΒ)] L n f -f =O ((Α(n )n )Β).推论3　若Α(n )=O (n 1-Ε)(0<Ε≤1),0<Β<1,且f ∈C [0,1],则L n f -f =O (n -ΕΒ)]Ξ2Υ(f ,t )=O (t 2ΕΒ)Ξ(f ,t )=O (t ΕΒ).2　定理的证明定理1的证明　注意到[2]B n f -f ≤M (Ξ2Υ(f ,1n)+1nf ),f ∈C [0,1],则得 L n f -f ≤ B n f -f + B n f -L n f ≤M (Ξ2Υ(f ,1n)+1nf )+∑nk =0pn ,k( ) f (k n )-f (kn +Α(n )) ≤M (Ξ2Υ(f ,1n)+Ξ(f ,Α(n )n)+1nf ).□定理2的证明基于以下若干引理.首先,注意到 f (k n +Α(n )) ≤ f (0≤k ≤n ),且以f (k n +Α(n ))来代替f (kn),则类似于B ern stein 算子的估计[4],直接计算可得引理1　下列不等式成立 L ′n f ≤2n f (f ∈C [0,1]), L ″n f ≤4n 2f (f ∈C [0,1]),L ′n f ≤ f ′ (f ∈C 1[0,1]), L ″n f ≤ f ″(f ∈C 2[0,1]), Υ2L ″n f ≤2n f (f ∈C [0,1]).引理2　设f ∈C 2[0,1],则 Υ2L ″n f ≤ Υ2f ″+1nf ″ .证明　记h =1n +Α(n ),则L ″n (f ;x )=n (n -1)∑n -2k =0pn -2,k(x )∃2k f (k +1n +Α(n )),于是 Υ2(x )L ″n(f ;x ) =∑n -2k =0(k +1)(n -k -1)p n ,k +1(x )∃2h f (k +1n +Α(n ))=h-2∑n -1k =1kn +Α(n ) n -k n +Α(n )p n ,k (x ) ∃2h f (k n +Α(n )) ,所以Υ2(x )L ″n (f ;x ) ≤h-2∑n -1k =1Υ2(k n +Α(n ))p n ,k (x ) ∃2h f (k n +Α(n )) .令y =k n +Α(n )(1≤k ≤n -1),则h ≤y ≤1-h ,而对于 u ≤h ,有 1-2y -u ≤1.于是Υ2(y )= Υ2(y +u )-u (1-2y -u ) ≤Υ2(y +u )+ u ≤Υ2(y +u )+h ,从而 Υ2(y ) ∃2h f (y ) ≤Υ2(y )∫h 2-h2∫h2-h2f ″(y +s +t )d s d t≤∫h2-h2∫h2-h2(Υ2(y +s +t )+h ) f ″(y +s +t ) d s d t ≤h 2(Υ2f ″ +1nf ″ ),即有 Υ2L ″n f ≤ Υ2f ″+1nf ″ .其次,需要以下两个有关非负数列的有趣结果.引理3[5]　设{Λn },{Τn },{Ωn }均为非负数列,且Λ1=Τ1=0,若不等式(0<r <s ,1≤k ≤n )Λn ≤(k n )r Λk +Τk +Ωk ,Τn ≤(k n)sΤk +Ωk 对于n ∈N 成立,则有Λn ≤M n-1∑nk =1kr -1Ωk .引理4　设{Τn },{Ωn }为非负数列,s >0,若Τn ≤(k n)sΤk +M Ωk ,1≤k ≤n ,则Τn ≤M s n-s(∑nk =1ks -1Ωk +Τ1),其中M s 表示与s 有关的正常数.证明　文献[5]之引理2.1对Τ1=0与M =1的情形给出了证明,采用与其相似的方法容易证明引理4.现在来证明定理2.令Λn =n -1 Υ2L ″n f ,Τn =n -2 L ″n f ,Ωn =4 L n f -f .显然,Λ1=Τ1=0.对于1≤k ≤n ,因为 Λn ≤n -1 Υ2L ″n L k f +n -1Υ2L ″n (L k f -f )≤n -1 Υ2L ″k f +n -2 L ″k f +2 L k f -f≤kn Λk +Τk +Ωk , Τn ≤n -2 L ″n L k f +n -2 L ″n (L k f -f )≤n -2 L ″k f +4 L k f -f =(k n)2Τk +Ωk ,此处用到了引理1,引理2.所以,由引理3得Λn ≤M n-1∑nk =1Ωk,即有Υ2L ″n f ≤M∑nk =1Lkf -f .(2.1) 若对Τn =n -1 L ′n f ,Ωn 意义如上,则由引理1得 Τn ≤n -1 L ′n L k f +n -1 L ′n (L k f -f ) ≤n -1 L ′k f +4 L k f -f=knΤk +Ωk ,1≤k ≤n ,故由引理4得Τn ≤M n-1(∑nk =1Ωk+Τ1),即有 L ′n f ≤M (∑n k =1Lkf -f + L′1f )≤M (∑nk =1Lkf -f + f ).(2.2)对于n ≥2,存在m ∈N ,使得n2≤m ≤n ,且 L m f -f ≤ L k f -f (n2≤k ≤n ),则 L m f -f ≤4n∑n k =1L k f -f .(2.3)于是,由(1.1),(2.1)及(2.3)得 Ξ2Υ(f ,1n)≤M K 2Υ(f ,1n)≤M ( L m f -f +1nΥ2L ″m f )　≤Mn(∑nk =1Lkf -f +∑mk =1Lkf -f )≤M n-1∑nk =1Lkf -f .而由(1.2),(2.2),(2.3)得 Ξ(f ,1n)≤M K (f ,1n)≤M ( L m f -f +1nL ′m f )≤M n -1(∑nk =1Lkf -f + f ).□参　考　文　献[1]　李　松.Bern stein 2Sikkem a 算子的正逆定理[J ].应用数学学报,1996,19(1):144—148.[2]　D itzian Z and To tik V .M od u li of Sm oothness [M ].Sp ringer 2V erlag ,N ew Yo rk Berlin H eidelberg ,1987.[3]　Sikkem a P C .Uβber d ie S chu rerschen L inea ren P ositiven Op era toren [J ].Indag .M ath .,1975,37:243-253.[4]　Beren s H and L o ren tz G G .Inverse theore m s f or B ernstein p oly no m ia ls [J ].Indiana U n iv .M ath .J .,1972,21(8):693-708.[5]　E rich van W ickeren .S teck in 2M a rchaud 2Typ e inequa lities in connection w ith B ernstein p oly no m ia ls [J ].Con str .A pp rox .,1986,2:331-337.[6]　王仁宏.无界函数逼近[M ].北京:科学出版社,1983.Approx i m ation of Bern ste i n -Sikkema Operators X iong Q ing liang Cao F eilong(Shaoxing L u Xun ′s Co llege ,Zhejiang 312000)　(D ep t of M ath .,N ingxia U niv .,Yinchuan 750021)AbstractT he p rob lem s of app rox i m ati on fo r B ern stein 2Sikkem a op erato rs are studied ,and the app rox i m ati on strong 2typ e direct theo rem and w eak 2typ e inverse theo rem are ob tained .T he resu lts of the reference [1]are i m p roved .Keywords B ern stein 2Sikkem a op erato rs ,m odu lu s of s m oo thness ,app rox i m ati on .。