Gaussian filtering Fourier multiplier (${\hat{G}}_{\sigma }(\xi )=e^{-{\sigma }^2|\xi {|}^2/2}$)

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.