Upon Fourier transformation the convolution becomes a product,
$${\cal F}[\mu_{C}*f](k) = \exp\left(-\tfrac{1}{2}\sum_{n,m}k_n C_{nm} k_m\right){\cal F}[f](k).$$
Inverse Fourier transformation, with $k_n=i\partial/\partial x_n$, then gives Equation (1).