  Over the last few years, needlets have a emerged as a useful tool for the
analysis of Cosmic Microwave Background (CMB) data. Our aim in this paper is
first to introduce in the CMB literature a different form of needlets, known as
Mexican needlets, first discussed in the mathematical literature by Geller and
Mayeli (2009a,b). We then proceed with an extensive study of the properties of
both standard and Mexican needlets; these properties depend on some parameters
which can be tuned in order to optimize the performance for a given
application. Our second aim in this paper is then to give practical advice on
how to adjust these parameters in order to achieve the best properties for a
given problem in CMB data analysis. In particular we investigate localization
properties in real and harmonic spaces and propose a recipe on how to quantify
the influence of galactic and point source masks on the needlet coefficients.
We also show that for certain parameter values, the Mexican needlets provide a
close approximation to the Spherical Mexican Hat Wavelets (whence their name),
with some advantages concerning their numerical implementation and the
derivation of their statistical properties.
