L_p范数下多元正系数代数多项式倒数对非负连续函数的逼近  

L_p approximation of non-negative continuous functions by reciprocals of multivariate algebraic polynomials with postive coefficients

作　　者：盛宝怀[1] 周颂平[2] 

机构地区：[1]绍兴文理学院数学系 [2]浙江工程学院数学研究所,浙江杭州310018

出　　处：《高校应用数学学报（A辑）》2003年第4期455-466,共12页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(10141001);宁波市博士基金;宁波大学博士后基金(02J20102-06);浙江省自然科学基金 (1 0 2 0 0 2 )

摘　　要：设d≥1为正整数,S为Rd中的单纯形,C(S)为S上连续函数类,f(x)∈C(S),f(x)≥0,f(x) 0,p>1,‖.‖p为通常的Lp范数,‖.‖为一致范数,则存在Pn(x)∈Π+n,d={Pn(x):Pn(x)= |k|≤nakxk(1-|x|)n-|k|:x∈S,ak≥0},常数C>0使‖f-1Pn‖p≤Cω2φf,14n+‖f‖n,这里对k,x∈Rd,k=(k1,k2,…,kd),x=(x1,x2,…,xd),记|k|=k1+k2+…+kd,|x|=x1+x2+…+xd,xk=xk11xk22…xkdd,ω2φ(f,t)为单纯形S上关于一致范数的二阶Ditzian-Totik光滑模.Let d≥1 be a positive integer, S be a simplex in R\+d,C(S) be the class of continuous functions defined on S,f(x)∈C(S),f(x)≥0,f(x)0,p>1,‖·‖\-p be the ordinary L\-p norm,‖·‖ be the uniform norm.It is proved in the present paper that there exist P\-n(x)∈Π +n,d =Pn(x):P\-n(x)=|k|≤na\-kx\+k(1-|x|) n-|k| :x∈S,a\-k≥0 and a constant C>0 such that ‖f-1P\-n‖\-p≤Cω 2φf,14n+‖f‖n. Here,for k,x∈R\+d,k=(k\-1,k\-2,...,k\-d),x=(x\-1,x\-2,...,x\-d), set |k|=k\-1+k\-2+...+k\-d,|x|=x\-1+x\-2+...+x\-d,x\+k=x k1 1x k2 2...x kd d, and ω 2φ(f,t) is the second order Ditzian\|Totik continuous modulus defined on S .

关 键 词：多项式倒数逼近 非负连续函数 单纯形 Bernstein—Durrmeyer多项式算子 Ditzian—Totik光滑模 LP范数 

分 类 号：O174.42[理学—数学]