周期可微函数类和卷积函数类在L_q尺度下的相关n-宽度  

Relative n-Widths of the Classes of Differentiable Functions and the Periodic Convolution Classes in L_q Metric

作　　者：刘永平[1] 杨唯[2] 

机构地区：[1]北京师范大学数学科学学院数学与复杂系统教育部重点实验室,北京100875 [2]东北师范大学数学与统计学院,长春130024

出　　处：《数学学报（中文版）》2010年第6期1087-1096,共10页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(10771016)

摘　　要：设X_j∈R,j=1,2,…,r,是具有实零点的r次实系数代数多项式.称之为由P_r(x)导出的微分算子,其中D=d/dx且I是单位算子.本文涉及的函数类W_p^(P_r)由所有在T=[-π,π]上有(r-1)阶绝对连续导数,r阶导数属于L_p(T)且满足‖P_r(D)f‖_p≤1的函数f组成.当P_r(D)=D^r,W_p^(P_r)是通常的Sobolev类W_p^r.本文研究了函数类W_∞^(P_r)和W_1^(P_r)在L_q(T)尺度下的相关n宽度,得到这些类上相关宽度K_n(W_∞^(P_r),W_∞^(P_r))_q和K_n(W_1^(P_r),W_1^(P_r))_q的渐近估计.上述结果中去掉了在杨连红和刘永平同类文章中考虑的微分算子的共轭性条件.此外,本文也考虑了周期卷积类M_p^G在p=1和∞时的同类问题,其中卷积核为PF密度的周期化,得到相关宽度K_n(W_p^G,M_p^G)_1在p=1和∞时的渐近结果.Let,x_j∈R,j=1,2,…,r,be a real coefficient algebraic polynomial of order r with real roots.Then is called to be the differential operator determined by P_r（x）,where D =d/dx and I is the identical operator.The class W_p^（P_r） concerned in this paper is consisted of functions f∈L_p（T） whose（r-1l）-st derivative is absolutely continuous on T =[-π,π],r-th derivative belongs to L_p（T） and satisfies‖P_r（D）f‖_p≤1.When P_r（D） = D^r,W_p^（P_r） is the usual Sobolev class W_p^r.In the present paper,the relative n-widths of W_∞^（P_r） and W_1^（P_r） in L_q（T） metric are studied,and the asymptotic estimates of K_n（W_∞^（P_r）,W_∞^（P_r）_q and K_n（W_1^（P_r）,W_1^（P_r）_q are obtained.In our results,we removed the self-conjugate condition of the differential operator considered in Yang Lianhong and Liu Yongping＇s paper. Furthermore,we consider the periodic convolution classes W_p^G,p = 1 and∞,the convolution kernels of which are the periodization of PF densities.By the analysis of the kernels,we obtain the asymptotic estimates of K_n（W_p^G,W_p^G）,p = 1 and∞.

关 键 词：相关宽度 可微函数类 周期卷积类 

分 类 号：O174[理学—数学]