Fourier Transform of Dirac Delta Distribution

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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:

\(\ds \map {\hat \delta} \phi\) \(=\) \(\ds \map \delta {\hat \phi}\) Definition of Fourier Transform of Tempered Distribution
\(\ds \) \(=\) \(\ds \map {\hat \phi} 0\) Definition of Tempered Dirac Delta Distribution
\(\ds \) \(=\) \(\ds \int_{- \infty}^\infty \map \phi x e^{-2 \pi i 0 x} \rd x\) Definition of Fourier Transform of Real Function
\(\ds \) \(=\) \(\ds \int_{- \infty}^\infty \map \phi x \cdot 1 \rd x\)
\(\ds \) \(=\) \(\ds \map {\mathbf 1} \phi\)

$\blacksquare$

Sources