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

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

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

<math>f(x_n) = y_n<math>.

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

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

with

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

where

<math>\omega_N^{i}<math>