Dirac distribution

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

It is closely related to Dirac measures, in the language of measure theory.

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

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

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 of the delta-distribution)**

The Fourier transform (this def.) of the delta distribution, via this example, is the constant function on 1:

$\begin{aligned}
\widehat {\delta}(k)
& =
\underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- 2\pi i k x} \, d x
\\
& =
1
\end{aligned}$

This implies by the Fourier inversion theorem (this prop.) that the delta distribution itself has equivalently, in generalized function-notation, the expression

(1)$\begin{aligned}
\delta(x)
& =
\widehat{\widehat{\delta}}(-x)
\\
& =
\int_{k \in \mathbb{R}^n} e^{2 \pi i k \cdot x} \, d k
\end{aligned}$

in the sense that for every function with rapidly decreasing partial derivatives $f \in \mathcal{S}(\mathbb{R}^n)$ we have

$\begin{aligned}
f(x)
& =
\underset{y \in \mathbb{R}^n}{\int}
f(y) \delta(y-x) \, dvol(y)
\\
& =
\underset{y \in \mathbb{R}^n}{\int}
\underset{k \in \mathbb{R}^n}{\int}
f(y) e^{2 \pi i k \cdot (y-x)}
\, dvol(k)\, dvol(y)
\\
& =
\underset{k \in \mathbb{R}^n}{\int}
e^{- 2 \pi i k \cdot x}
\underset{= (-1)^n\widehat{f}(-k)}{
\underbrace{
\underset{y \in \mathbb{R}^n}{\int}
f(y) e^{2 \pi i k \cdot y}
\, dvol(y)
}
}
\, dvol(k)
\\
& =
\widehat{\widehat{f}}(-x)
\end{aligned}$

which is just the statement of the Fourier inversion theorem for smooth functions (this prop.).

It is clear that:

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

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 $u$.

(e.g. Hörmander 90, theorem 2.3.4)

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

Last revised on October 26, 2019 at 10:25:22. See the history of this page for a list of all contributions to it.