Dirichlet conditions

We state Dirichlet's theorem assuming f is a periodic function of period 2π with Fourier series expansion where

${\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.}$

The analogous statement holds irrespective of what the period of f is, or which version of the Fourier expansion is chosen (see Fourier series).

Dirichlet's theorem: If f satisfies Dirichlet conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{inx}={\frac {f(x^{+})+f(x^{-})}{2}},}$

where the notation

${\displaystyle f(x^{+})=\lim _{y\to x^{+}}f(y)}$

${\displaystyle f(x^{-})=\lim _{y\to x^{-}}f(y)}$

denotes the right/left limits of f.

A function satisfying Dirichlet's conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where f is continuous,

${\displaystyle {\frac {f(x^{+})+f(x^{-})}{2}}=f(x)}$

Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier series for f converges and is equal to f wherever f is continuous.

• "Dirichlet conditions". PlanetMath.