Distributional Derivative of Heaviside Step Function
Jump to navigation
Jump to search
Theorem
Let $H : \R \to \closedint 0 1$ be the Heaviside step function.
Let $T \in \map {\DD'} \R$ be a distribution corresponding to $H$.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then the distributional derivative of $T$ is $\delta$.
Proof
Let $\phi \in \map \DD \R$ be a test function.
Then:
\(\ds \map {T'} \phi\) | \(=\) | \(\ds -\map T {\phi'}\) | Definition of Distributional Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_{-\infty}^\infty \map H x \map {\phi'} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int_0^\infty \map {\phi'} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi 0 - \map \phi \infty\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi 0\) | Definition of Test Function, Value of Compactly Supported Function outside its Support, $\infty$ is not in a bounded subset of real numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \delta \phi\) | Definition of Dirac Delta Distribution |
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense