nLab
Dirac distribution

Contents

Idea

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

Properties

Proposition

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

Θ=δ. \partial \Theta = \delta \,.
Proof

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

Θ(x)b(x)dx =Θ(x)b(x)dx = 0 b(x)dx =(b(x)| xb(0)) =b(0) =δ(x)b(x)dx. \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 n\mathbb{R}^n is

δ^(k)=δ(x)e ixkd nx \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

δ(x)=(2π) ne ixkd nk. \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 uu with supp(u)={p}supp(u) = \{p\} is a finite sum of multiplies of derivatives of the delta distribution at that point:

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

where {c α} α\{c^\alpha \in \mathbb{R}\}_\alpha, and for kk \in \mathbb{N} the order of uu.

(e.g. Hörmander 90, theorem 2.3.4)

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)