Improving the full spectrum fitting method: accurate convolution with
  Gauss-Hermite functions
  I start by providing an updated summary of the penalized pixel-fitting (pPXF)
method, which is used to extract the stellar and gas kinematics, as well as the
stellar population of galaxies, via full spectrum fitting. I then focus on the
problem of extracting the kinematic when the velocity dispersion $\sigma$ is
smaller than the velocity sampling $\Delta V$, which is generally, by design,
close to the instrumental dispersion $\sigma_{\rm inst}$. The standard approach
consists of convolving templates with a discretized kernel, while fitting for
its parameters. This is obviously very inaccurate when $\sigma<\Delta V/2$, due
to undersampling. Oversampling can prevent this, but it has drawbacks. Here I
present a more accurate and efficient alternative. It avoids the evaluation of
the under-sampled kernel, and instead directly computes its well-sampled
analytic Fourier transform, for use with the convolution theorem. A simple
analytic transform exists when the kernel is described by the popular
Gauss-Hermite parametrization (which includes the Gaussian as special case) for
the line-of-sight velocity distribution. I describe how this idea was
implemented in a significant upgrade to the publicly available pPXF software.
The key advantage of the new approach is that it provides accurate velocities
regardless of $\sigma$. This is important e.g. for spectroscopic surveys
targeting galaxies with $\sigma\ll\sigma_{\rm inst}$, for galaxy redshift
determinations, or for measuring line-of-sight velocities of individual stars.
The proposed method could also be used to fix Gaussian convolution algorithms
used in today's popular software packages.
