Fourier series

Diriclet 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.