Fourier Transform of Dirac Delta Distribution

Theorem
Let $\delta \in \map {\SS'} \R$ be the Dirac delta distribution.

Let $\mathbf 1 : \map \SS \R \to \R$ be the constant tempered distribution such that for all $\phi \in \map \SS \R$ we have:


 * $\ds \map {\mathbf 1} \phi = \int_{-\infty}^\infty 1 \cdot \map \phi x \rd x$

Then in the distributional sense it holds that:


 * $\hat \delta = \mathbf 1$

where the hat denotes the Fourier transform of a tempered distribution.

Theorem
Let $\phi \in \map \SS \R$ be a Schwartz test function.

Then: