Gaussian filtering Fourier multiplier
= Gaussian filtering Fourier multiplier
{c}
{title2=$\widehat G_\sigma(\xi)=e^{-\sigma^2|\xi|^2/2}$}
The <Fourier transform of a Gaussian> and the <convolution theorem> give this multiplier with the angular-frequency convention $e^{-ix\cdot\xi}$. In cycles-per-length <frequencies> $k$, it is $e^{-2\pi^2\sigma^2|k|^2}$. Each coordinate has variance $\sigma^2$, and higher <frequencies> receive stronger attenuation.