Trigonometric interpolation

The approximate representation of a function $f$ 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

$$T(x)=\sum_{k=0}^{2n}y_kt_k(x),\label{*}\tag{*}$$

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 \eqref{*} 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