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
m
m
-order differentiable function
f
f
on [
0
,
1
0,1
], we will construct an
m
m
-degree algebraic polynomial
P
m
{P}_{m}
depending on values of
f
f
and its derivatives at ends of [
0
,
1
0,1
] such that the Fourier coefficients of
R
m
=
f
−
P
m
{R}_{m}=f-{P}_{m}
decay fast. Since the partial sum of Fourier series
R
m
{R}_{m}
is a trigonometric polynomial, we can reconstruct the function
f
f
well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.