关于光滑函数类上的最优取样

Concerning an Optimal Sampling Problem for Some Classes of Smooth Functions

§1.问题的提出 设X_1,X_2,X_3是实或复的线性赋范空间,KX_1为零点对称凸集,S是X_1→X_2的线性算子.I是X_1→X_3的线性算子。任取x∈K,Ix称为x的信息,I是K的一信息算子。根据[1]。

In this note we consider optimal recovery problem for some classes of functions which are defined by a linear integral operator and are embeded in the space Lq,1≤ q≤ + ∝. A set of sampling points is any subset of the domain of definition for the function class. Over the set of all sets of sampling points whose cardinal≤N we prove that the N-th minimal diameter of information for the class of functions under consideration is bounded below by the Gelfand N-width of the function class in Lq. It is pointed out that in some special cases the inequalities turn into equalities, and the optimal sampling point set may be accurately identified. Among other things the equality is established for the Sobolev class Wpr(r≥1,1≤p≤q≤+∞), Wpr, as well as for some periodic convolution class whose kernel is CVD, etc.