Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets

TL;DR

This paper introduces Mexican needlets for CMB analysis, compares them with standard needlets, and provides practical guidelines for parameter tuning to optimize their performance in real-world applications.

Contribution

It is the first to incorporate Mexican needlets into CMB analysis and offers a detailed comparison and parameter tuning strategies for practical use.

Findings

Mexican needlets closely approximate Spherical Mexican Hat Wavelets for certain parameters.

Guidelines are provided for tuning needlet parameters to optimize localization and mask influence.

Mexican needlets offer advantages in numerical implementation and statistical analysis.

Abstract

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.