Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Abstract

Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an mm-order differentiable function ff on [0,10,1], we will construct an mm-degree algebraic polynomial Pm{P}_{m} depending on values of ff and its derivatives at ends of [0,10,1] such that the Fourier coefficients of Rm=f−Pm{R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series Rm{R}_{m} is a trigonometric polynomial, we can reconstruct the function ff well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

Keywords

• fourier approximation
• trigonometric polynomial
• differential equations
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