Knowlpedia

Definition

Dirac delta distribution

The distribution that evaluates a test function at one specified point.

For a∈Rna\in\mathbb R^n, the Dirac delta distribution at aa is

⟨δa,φ⟩=φ(a)\langle\delta_a,\varphi\rangle=\varphi(a)

for every φ\varphi. Evaluation is linear and continuous in the test-function topology, so this defines a distribution.

Point mass and derivatives

It is also the distribution associated with the unit point-mass measure and extends continuously to Schwartz functions, hence is tempered. Its derivative is defined by ⟨∂jδa,φ⟩=−∂jφ(a)\langle\partial_j\delta_a,\varphi\rangle=-\partial_j\varphi(a). The delta is not represented by a locally integrable function. With the Fourier convention e−2πix⋅ξe^{-2\pi ix\cdot\xi}, δ0^=1\widehat{\delta_0}=1 as tempered distributions.