Then the Fourier transform of the Dirac distribution is
\[ \widehat{\delta}_0\equiv 1. \]
Proof
Let \( \varphi \in \mathcal{S}(\mathbb{R}^n) \). Then
\[ \hat{\varphi}(s) = \mathcal{F}(\varphi)(s) = \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} \varphi(x) e^{-is \cdot x} \, dx. \]And therefore
\[ \hat{\delta}_0(\varphi) \approx \int_{\mathbb{R}^n} \varphi(x) \, dx = 1(\varphi). \]