Chemistry - Trigonometric interpolation

In the mathematical subfield of numerical analysis, trigonometric interpolation is a special form of interpolation on the unit circle in the complex plane using trigonometric polynomials.

Complex trigonometric interpolation

Given N real numbers of the form

$x_n = \frac{2 \pi n}{N} \mbox{ , } n = 0,\ldots,N-1$

and N complex numbers y_n with n = 0,...,N-1 we are trying to find a function f with

f (x n) = y n

Due to the Stone-Weierstrass theorem this function exists and is unique. It is called complex trigonometric polynomial of degree N-1 and has the form

$T_{N-1}(x) = \sum_{n=0}^{N-1} a_n e^{\mathrm{i}nx}$

with

$a_n = \frac{1}{N} \sum_{m=0}^{N-1} y_n \omega_{N}^{-mn} \mbox{ , } m = 0,\ldots,N-1$

where

$\omega_N^{i}$