Wavelets on Manifolds and Statistical Applications to Cosmology

Abstract

We outline how many of the pioneering ideas that were so effective in developing wavelet theory on the real line, can be adapted to the manifold setting. In this setting, however, arguments using Fourier series are replaced by methods from modern harmonic analysis, pseudodifferential operators and PDE. We explain how to construct nearly tight frames on any smooth, compact Riemannian manifold, which are well-localized both in frequency (as measured by the Laplace–Beltrami operator) and in space. These frames can be used to characterize Besov spaces on the manifold for the full range of indices, in analogy to the Frazier–Jawerth result on the real line. We explain how our methods can be extended beyond the study of functions, to the wavelet analysis of sections of a particular line bundle on the sphere, which is important for the analysis of the polarization of CMB(cosmic microwave background radiation). The wavelet approach to CMB has been advocated by many people, including our frequent collaborators, the statistician Domenico Marinucci, and the physicist Frode Hansen, who earlier used spherical needlets to study CMB temperature.