The best quadrature based on given Hermite information for the Sobolev class KW^r[a,b]

作     者：WANG Xinghua & YANG Shijun Department of Mathematics, Zhejiang University, Hangzhou 310028, China Department of Mathematics, Hangzhou Normal College, Hangzhou 310036, China 

作者机构：Department of Mathematics Zhejiang University Hangzhou China Department of Mathematics Hangzhou Normal College Hangzhou China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2006年第49卷第8期

页      面：1146-1152页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：This work was supported by the National Natural Science Foundation of China (Grant No. 10471128) 

主　　题：Sobolev class, Hermite information, perfect spline, optimal recovery, best quadrature. 

摘      要：As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. In [a,b]. For a function f ∈ KWr[a,b], its values and derivatives up to r-1 order at a set of nodes x are known. These values are said to be the given Hermite *** work reports the results on the best quadrature based on the given Hermite information for the class KWr[a,b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product,the best interpolation formula for the class KWr[a,b] is also obtained.