In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f(x) to be equal to 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 (it is the midpoint of the values of the discontinuity). These conditions are named after Johann Peter Gustav Lejeune Dirichlet.
The conditions are:
- f(x) must be absolutely integrable over a period.
- 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 bounded
The last three conditions are satisfied if f is a function of bounded variation.
Read more about Dirichlet Conditions: Dirichlet's Theorem For 1-Dimensional Fourier Series
Famous quotes containing the word conditions:
“The worst of my actions or conditions seem not so ugly unto me as I find it both ugly and base not to dare to avouch for them.”
—Michel de Montaigne (1533–1592)