INTRODUCING MEXICAN NEEDLETS FOR CMB ANALYSIS: ISSUES FOR PRACTICAL APPLICATIONS AND COMPARISON WITH STANDARD NEEDLETS

Over the last few years, needlets have emerged as a useful tool for the analysis of cosmic microwave background (CMB) data. Our aim in this paper is first to introduce into the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller and Mayeli. 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 for WMAP and Planck data 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 space and propose a recipe for quantifying 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 derivation of their statistical properties.