Decomposition and approximation of multivariate functions on the cube

Abstract

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 Σ j ϕ j ψ j , where each ϕ j can be extended to a smooth periodic function, each ψ j is an algebraic polynomial, and each φ j ψ j 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.