  I present a method for the fast convolution of a model galaxy profile by a
point-spread function (PSF) model represented as a pixel grid. The method
relies upon three observations: First, most simple radial galaxy profiles of
common interest (deVaucouleurs, exponential, Sersic) can be approximated as
mixtures of Gaussians. Second, the Fourier transform of a Gaussian is a
Gaussian, thus the Fourier transform of a mixture-of-Gausssian approximation of
a galaxy can be directly evaluated as a mixture of Gaussians in Fourier space.
Third, if a mixture component would result in Fourier-space aliasing, that
component can be evaluated in real space. For mixture components to be
evaluated in Fourier space, we use the FFT for the PSF model, direct evaluation
of the Fourier transform for the galaxy, and the inverse-FFT to return the
result to pixel space. For mixture components to be evaluated in real
space---which only happens when the mixture components is much larger than the
PSF---we use a simple Gaussian approximation of the PSF, perform the
convolution analytically, and evaluate in real pixel space. The method is fast
and exact (to the extent that the mixture-of-Gaussians approximation of the
galaxy profile is exact) as long as the pixelized PSF model is well sampled.
This Fourier method can be seen as a way of applying a perfect low-pass filter
to the (typically strongly undersampled) galaxy profile before convolution by
the PSF, at exactly the Nyquist frequency of the PSF pixel model grid. In this
way, it avoids the computational expense of a traditional super-resolution
approach. This method allows the efficient use of pixelized PSF models (ie, a
PSF represented as a grid of pixel values) in galaxy forward model-fitting
approaches such as the Tractor.
