nLab Dirac distribution

Idea

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.

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

Example

(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.).

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 $u$.

(e.g. Hörmander 90, theorem 2.3.4)

Related entries

• Green's function

• Kronecker delta

• Dirac measure

References

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