Definition:Dirichlet Conditions

Definition

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:

\((\text D 1)\)	$:$							$f$ is absolutely integrable
\((\text D 2)\)	$:$							$f$ has a finite number of local maxima and local minima
\((\text D 3)\)	$:$							$f$ has a finite number of discontinuities, all of them finite

Examples

Reciprocal of $4 - x^2$

The function:

$\map f x = \dfrac 1 {4 - x^2}$

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

Sine of $\dfrac 1 {x - 1}$

The function:

$\map f x = \map \sin {\dfrac 1 {x - 1} }$

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 2$. Fourier Series