Fourier transform - Numerical Analysis

from pylab import *

1Fourier series¶

Let $f : [-\pi,\pi] \to \re$ be a function which is extended periodically to the whole real line. The Fourier series is defined as

Sf(x) = a_0 + \sum_{k=1}^\infty [ a_k \cos(k x) + b_k \sin(k x) ]

where

\begin{align*} a_0 &= \int_{-\pi}^\pi f(x) \ud x \\ a_k &= \int_{-\pi}^\pi f(x) \cos(k x) \ud x, \qquad k=1, 2, \ldots \\ b_k &= \int_{-\pi}^\pi f(x) \sin(k x) \ud x, \qquad k=1, 2, \ldots \end{align*}