Band-Limited Localized Parseval Frames and Besov Spaces on Compact Homogeneous Manifolds

Abstract

Abstract. In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (=band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-known ϕ-transform on Euclidean spaces.

Splines and Wavelets on Geophysically Relevant Manifolds

“…The following theorem holds for any compact manifold [24], [51]. , there exist strictly positive coefficients µ ν > 0 for which the following equality holds for all functions in E ω (M ):…”

“…This implies in particular that the corresponding operator on The following important result was obtained in [24], [51].…”

“…The goal of the present study is to describe new constructions and applications of splines and bandlimited and localized frames in a space L 2 (M ), where M is a compact Riemannian manifold. Our article is a summary of some results for compact manifolds that were obtained in [5], [6], [24], [46]- [55]. To the best of our knowledge these are the papers which contain the most general results about splines, frames and Shannon sampling on compact Riemannian manifolds along with applications to Radon-type transforms on manifolds.…”

“…More details about the Radon transform on general homogeneous manifolds and in particular on SO(3) can be found in [30] and in [4]- [9], [32], [33], [56]. In section 2 we formulate two our imprtant results (see [24] and [51]): a Theorem 2.1 about positive cubature formulas on general compact Riemnnian manifolds and Theorem 2.2 about eigenfunctions of a Casimir operator on a compact homogeneous manifold.…”

“…In section 4 we construct bandlimited and localized Parseval frames in L 2 (M ) where M is a homogeneous manifold [24], [51].…”