Improving the full spectrum fitting method: accurate convolution with Gauss–Hermite functions

Abstract

I start by providing an updated summary of the penalized pixel-fitting (PPXF) method that 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 kinematics when the velocity dispersion is smaller than the velocity sampling V that is generally, by design, close to the instrumental dispersion inst . The standard approach consists of convolving templates with a discretized kernel, while fitting for its parameters. This is obviously very inaccurate when 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 undersampled 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 . This is important e.g. for spectroscopic surveys targeting galaxies with 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.