Dirichlet condition

This article explains a convergence condition for Fourier series; for the boundary condition in differential equations; see Dirichlet boundary condition .

The Dirichlet condition , also called Dirichlet's theorem , is named after Peter Gustav Lejeune Dirichlet and specifies when the Fourier series converges point by point to the output function.

statement

Let be a function defined in the interval that fulfills the following properties: ${\displaystyle f}$ ${\displaystyle [-T/2,T/2]}$

1. The interval can be broken down into a finite number of sub-intervals in which is continuous and monotonic. ${\displaystyle [-T/2,T/2]}$ ${\displaystyle f}$
2. The (finitely many) points of discontinuity are all of the first kind , that is, there are right and left-hand limit values , and . ${\displaystyle f(t_{0}+)}$ ${\displaystyle f({t_{0}}-)}$

Then the Fourier series converges in each to ${\displaystyle t\in [-T/2,T/2]}$

${\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cdot \cos(n\omega t)+b_{n}\cdot \sin(n\omega t))={\begin{cases}f(t),&{\mbox{wenn }}f{\mbox{ in t stetig}}\\(f(t+)+f(t-))/2,&{\mbox{wenn }}f{\mbox{ in t unstetig}}\end{cases}}}$.

swell

• Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 .