Multiresolution Analysis on Compact Riemannian Manifolds

On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

“…A frame is said to be tight if c = C, in which case C is referred to as the tightness constant. Following [Pes13] and the references therein (see also [GM09b,GM09c,NPW06b]), our aim here is to build a frame {ψ j,k } on L 2 (M): fix Q ∈ N and consider the set H Q , given by the span of eigenfunctions u q such that q ≤ Q, also called the space of band-limited functions on M with bandwidth Q. Define an ǫ-lattice over M, given by a set of points {ξ k } on M, which can be viewed as the centers of balls such that:…”

“…Needlets on compact manifolds.This subsection is concerned with the construction of needlet systems over compact homogeneous manifolds, following [KNP12, GP11] (cf. also[Pes13]). Let (M, g) be a smooth compact homogeneous Riemannian manifold of dimension d, with no boundaries.…”

Adaptive global thresholding on the sphere

“…where in this case the set {ξ j,k , λ j,k } characterizes a suitable partition of M, given by a ε-lattice on M, with ε = λ j,k . Further details and technicalities concerning ε-lattices can be found in Pesenson [41]. Analogously to the spherical case, for f ∈ L 2 (M) and arbitrary j ≥ 0 and k ∈ {1, .…”

Splines and Wavelets on Geophysically Relevant Manifolds

“…In the last two decades, the importance of these and other applications triggered the development of various generalized wavelet bases and frames suitable for the unit spheres S 2 and S 3 and the rotation group of R 3 . Our list of references is very far from being complete [1]- [11], [13]- [28], [31]- [36], [40]- [55]. More references can be found in monographs [19], [20], [34].…”

“…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.…”