Dirichlet conditions

From Wikipedia, the free encyclopedia

Not to be confused with Dirichlet boundary condition.

In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f(x) to be equal the sum of its Fourier series at each point where f is continuous. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well. These conditions are named after Johann Peter Gustav Lejeune Dirichlet.

The conditions are:

f(x) must have a finite number of extrema in any given interval
f(x) must have a finite number of discontinuities in any given interval
f(x) must be absolutely integrable over a period.
f(x) must be bounded

[edit] Dirichlet's Theorem for 1-Dimensional Fourier Series

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

$f(x) \sim \sum_{n = -\infty}^\infty a_n e^{inx}$ ,

where

$a_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx}\, dx.$

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

$\sum_{n = -\infty}^\infty a_n e^{inx} = \frac{1}{2}(f(x+) + f(x-))$ ,

where the notation

$f(x+) = \lim_{y \to x^+} f(y)$

$f(x-) = \lim_{y \to x^-} f(y)$

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,

$\displaystyle f(x+) = f(x-) = f(x)$