Fourier series

Dirichlet Conditions

The conditions for a periodic function $f$ with period 2L to have a convergent Fourier series.

Theorem:

Let $f$ be a piecewise regular real-valued function defined on some interval [-L,L], such that $f$ has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to $f$ when $f$ is continuous and to the arithmetic mean of the left-handed and right-handed limit of $f$ at a point where it is discontinuous.

The Fourier Series

A Fourier series is an expansion of a periodic function $f$ in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.

$f(t)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k{e}^{\frac{j2\pi kt}{T}}$.

see also:Orthogonal Functions

Principle author of this page: Aric Goe