Decomposition and Approximation of Multivariate Functions on the Cube  

作　　者：Zhi Hua ZHANG 

机构地区：[1]College of Global Change and Earth System Science, Beijing Normal University

出　　处：《Acta Mathematica Sinica,English Series》2013年第1期119-136,共18页数学学报（英文版）

基　　金：Supported by Fundamental Research Funds for the Central Universities(Key Program);National Natural Science Foundation of China(Grant No.41076125);973 project(Grant No.2010CB950504);Polar Climate and Environment Key Laboratory

摘　　要：In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.

关 键 词：Decomposition of multivariate functions approximation of multivariate functions fundamental polynomial projection operator classification of boundary points 

分 类 号：O174[理学—数学]