Trigonometric interpolation
The approximate representation of a function in the form of a trigonometric polynomial
T(x)=A+\sum_{k=1}^n(a_k\cos kx+b_k\sin kx)
whose values coincide at prescribed points with the corresponding values of the function. Thus, it is always possible to choose the 2n+1 coefficients A, a_k, b_k, k=1,\dots,n, of the n-th order polynomial T so that its values are equal to the values y_k of the function at 2n+1 preassigned points x_k in the interval [0,2\pi). The polynomial has the form
T(x)=\sum_{k=0}^{2n}y_kt_k(x),\tag{*}
where
t_k(x)=\frac{\Delta x}{\Delta'(x)2\sin(x-x_k)/2},\quad\Delta(x)=\prod_{k=0}^{2n}2\sin\frac{x-x_k}{2}.
The polynomial T assumes an especially simple form in case the nodes x_k=2k\pi/(2n+1) are equi-distant; the coefficients are given by the formulas
A=\frac{1}{2n+1}\sum_{k=0}^{2n}y_k,
a_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\cos mx_k,
b_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\sin mx_k,\quad1\leq m\leq n.
Comments
The formula \ref{*} above for the trigonometric polynomial taking the prescribed values y_k at the nodes x_k is known as the Gauss formula of trigonometric interpolation, [a2].
References
[a1] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) |
[a2] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 29, 38 |
Trigonometric interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_interpolation&oldid=38916