Fourier series expansion

The Fourier series expansion is the process of representing a function $f \in L^{2} ([- π, π])$ as a sum of complex exponentials:

f (x) = \sum_{n \in Z} c_{n} e^{i n x},

where the coefficients $c_{n}$ are given by:

c_{n} = \frac{1}{2 π} \int_{- π}^{π} f (x) e^{- i n x} d x .

This expansion is valid in the $L^{2}$ sense, meaning the series converges to $f$ in the $L^{2}$ norm.