Geometric Space–Frequency Analysis on Manifolds

Abstract

This paper gives a survey of methods for the construction of spacefrequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections.After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley-Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley-Wiener frames in L 2 (M), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in L 2 (M), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on M.For compact Riemannian manifolds the theory extends to Lp and Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the socalled product property for eigenfunctions of certain operators and proves a cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay.Throughout the paper, the general theory is exemplified with the help of various concrete and relevant examples, such as the unit sphere and the Poincaré half plane.

Besov continuity for pseudo-differential operators on compact homogeneous manifolds

“…In connection with our comments in the introduction, a detailed description on euclidean models and the role of Besov spaces in the context of applications we refer the reader to the book of A. Cohen [9]. The reference Hairer, [19] explains the importance of the Besov spaces in the setting of the theory of regularity structures as well as a theorem of reconstruction and some interactions with stochastic partial differential equations; on the other hand, as it was pointed out in in [16] (see also references therein) several problems in signal analysis and information theory require non-euclidean models. These models include: spheres, projective spaces and general compact manifolds, hyperboloids and general non-compact symmetric spaces, and finally various Lie groups.…”

Density of sampling and interpolation in reproducing kernel Hilbert spaces

“…Currently there are numerous results about sampling theorems for 'sufficiently dense' sets. See, for instance, [15,29,[39][40][41]. These results need to be complemented by a critical density, provided that it exists at all.…”

Smooth Orthogonal Projections on Riemannian Manifold

“…More recently, Besov spaces on compact manifolds were studied by Geller and Mayeli [12]. A survey on a recent progress space-frequency analysis on compact Riemannian manifolds and Riemannian manifolds with bounded geometry can be found in [10]. In contrast to these developments, our results we do not require the assumption of bounded geometry.…”