ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES

作     者：T.Kilgore J.Szabados 

作者机构：Aubunn University U.S.A Mathematical Institute of the Hungarian of Academy of Sciences. Hungaria 

出 版 物：《Analysis in Theory and Applications》 (分析理论与应用（英文刊）)

年 卷 期：1994年第10卷第3期

页      面：93-103页

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

基　　金：Supported by International Research and Exchanges Board Supported by Hungarian National Science Foundation Grant No. 1910 

主　　题：ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES ITS 

摘      要：Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.