# Contents

## Idea

A Dirac distribution or Dirac $\delta$-distribution $\delta(p)$ is the distribution that is given by evaluating a function at a point $p$.

## Properties

###### Proposition

The distributional derivative of the Heaviside distribution $\Theta \in \mathcal{D}'(\mathbb{R})$ is the delta distribution $\delta \in \mathcal{D}'(\mathbb{R})$:

$\partial \Theta = \delta \,.$
###### Proof

For $b \in C^\infty_c(\mathbb{R})$ any bump function we compute:

\begin{aligned} \int \partial\Theta(x) b(x) \, d x & = - \int \Theta(x) \partial b(x)\, dx \\ & = - \int_0^\infty \partial b(x) d x \\ & = - \left( b(x)\vert_{x \to \infty} - b(0) \right) \\ & = b(0) \\ & = \int \delta(x) b(x) \, dx \,. \end{aligned}

### Fourier transform

###### Proposition

(Fourier transform)

The Fourier transform of the delta-distirbution on $\mathbb{R}^n$ is

$\widehat {\delta}(\vec k) \;=\; \int \delta(\vec x) e^{- i \vec x \cdot \vec k} d^n \vec x$

and hence the delta distribution itself has the expression

$\delta(\vec x) \;=\; (2 \pi)^{-n} \int e^{ i \vec x \cdot \vec k} d^n \vec k \,.$

### Relation to point-supported distributions

It is clear that:

The delta distribution is a compactly supported distribution, and in fact a point-supported distribution.

###### Proposition

Every point-supported distribution $u$ with $supp(u) = \{p\}$ is a finite sum of multiplies of derivatives of the delta distribution at that point:

$u = \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum} c^\alpha \partial_\alpha \delta(p)$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$, and for $k \in \mathbb{N}$ the order of $u$.

## References

• Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Revised on September 6, 2017 07:55:42 by Urs Schreiber (77.56.177.247)