Statistical Challenges in 21st Century Cosmology Proceedings IAU Symposium No. 306, 2014 A. F. Heavens, J.-L. Starck & A. Krone-Martins, eds. c  International Astronomical Union 2015 doi:10.1017/S1743921314010801 Radial 3D-Needlets on the Unit Ball Claudio Durastanti 1 , Yabebal T. Fantaye 2 , Frode K. Hansen 3 , Domenico Marinucci 4 and Isaac Z. Pesenson 5 1 Department of Mathematics, University of Tor Vergata, Rome, email: durastan@mat.uniroma2.it 2 Department of Mathematics, University of Tor Vergata, Rome, email: fantaye@mat.uniroma2.it 3 Institutt for teoretisk astrofysikk, University of Oslo, email: f.k.hansen@astro.uio.no 4 Department of Mathematics, University of Tor Vergata, Rome, email: marinucc@mat.uniroma2.it 5 Department of Mathematics, Temple University, Philadelphia, email: isaak.pesenson@temple.edu Abstract. We present a simple construction of spherical wavelets for the unit ball, which we label Radial 3D Needlets. We envisage an experimental framework where data are collected on concentric spheres with the same pixelization at different radial distances from the origin. The unit ball is hence viewed as a tensor product of the unit interval with the unit sphere: a set of eigenfunctions is therefore defined on the corresponding Laplacian operator. Wavelets are then constructed by a smooth convolution of the projectors defined by these eigenfunctions. Localization properties may be rigorously shown to hold in the real and harmonic domain, and an exact reconstruction formula holds; the system allows a very convenient computational implementation. Keywords. methods: data analysis, methods: statistical, cosmology: observations 1. Motivations and background It is well-known that Cosmology has recently experienced a golden era where datasets of unprecedented accuracy have become available, for instance on Cosmic Microwave Background radiation (see for instance Bobin et al. (2013), Planck XXIII (2013), and the references therein). These datasets are typically collected over the full-sky, usually covering thousands of square degrees, and hence data analysis methods based on flat sky approximations have become unsatisfactory; procedures which take into account the spherical nature of these observations have become mandatory. These methods are usually based on spherical Fourier analysis, and thus they are described in the frequency domain in terms of the spherical harmonics; this framework, however, can often turn out to be inadequate, due to the lack of localization properties in the real domain. Indeed, real data are typically characterized by huge regions of masked data and/or other features for which localization in the real domain is highly desirable; for this reason, several procedures involving spherical wavelets have become rather popular in astrophysical data analysis, see for instance McEwen et al. (2007), Starck et al. (2006), Donzelli et al. (2012) , Fa¨ y et al. (200), Marinucci et al. (2008), Pietrobon et al. (2008) and Starck et al. (2010) for a review. The next decade will probably be characterized by an equally amazing improvement on the quality of observational data: in particular three-dimensional investigation of 75 https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314010801 Downloaded from https://www.cambridge.org/core. IP address: 54.163.42.124, on 13 Jun 2020 at 03:22:27, subject to the Cambridge Core terms of use, available at