\begin{definition}[Definition:Dirichlet Conditions]
Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.
Let $\map f x$ be a real function which is defined and bounded on the interval $\openint \alpha \beta$.
The '''Dirichlet conditions''' on $f$ are sufficient conditions that $f$ must satisfy on $\openint \alpha \beta$ in order for:
:the Fourier series of $f$ at every $x$ in $\openint \alpha \beta$ to equal $\map f x$
:the behaviour of a Fourier series to be determined at finite discontinuities of $f$ in $\openint \alpha \beta$:
They are as follows:
{{begin-axiom}}
{{axiom | n = \text D 1
        | t = $f$ is absolutely integrable
}}
{{axiom | n = \text D 2
        | t = $f$ has a finite number of local maxima and local minima
}}
{{axiom | n = \text D 3
        | t = $f$ has a finite number of discontinuities, all of them finite
}}
{{end-axiom}}
\end{definition}