Definition:Dirichlet Conditions

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Definition

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.

Let $f \left({x}\right)$ be a real function which is defined and bounded on the interval $\left[{\alpha \,.\,.\, \beta}\right]$.


The Dirichlet conditions on $f$ are sufficient conditions that $f$ must satisfy on $\left[{\alpha \,.\,.\, \beta}\right]$ in order for:

the Fourier series of $f$ at every $x$ in $\left[{\alpha \,.\,.\, \beta}\right]$ to equal $f \left({x}\right)$
the behaviour of a Fourier series to be determined at finite discontinuities of $f$ in $\left[{\alpha \,.\,.\, \beta}\right]$:


They are as follows:

$(\mathrm D 1): \quad f$ is absolutely integrable.
$(\mathrm D 2): \quad f$ has a finite number of local maxima and local minima.
$(\mathrm D 3): \quad f$ 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:

$f \left({x}\right) = \sin \left({\dfrac 1 {x - 1} }\right)$

does not satisfy the Dirichlet conditions on the real interval $\left({0 \,.\,.\, 2 \pi}\right)$.



Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.


Sources