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 smoo...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.展开更多
基金Supported by Fundamental Research Funds for the Central Universities(Key Program)National Natural Science Foundation of China(Grant No.41076125)+1 种基金973 project(Grant No.2010CB950504)Polar Climate and Environment Key Laboratory
文摘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.
