Parseval wavelet frames on Riemannian manifold

TL;DR

This paper constructs Parseval wavelet frames on Riemannian manifolds, enabling analysis in various function spaces and extending wavelet theory to curved spaces with bounded geometry.

Contribution

It introduces a method to build Parseval wavelet frames on general Riemannian manifolds and characterizes function spaces via wavelet coefficients.

Findings

Existence of wavelet unconditional frames in L^p(M) for 1<p<∞.

Characterization of Triebel-Lizorkin and Besov spaces on compact manifolds.

Boundedness of Hestenes operators on manifolds with bounded geometry.

Abstract

We construct Parseval wavelet frames in $L^2(M)$ for a general Riemannian manifold $M$ and we show the existence of wavelet unconditional frames in $L^p(M)$ for $1 < p <\infty$. This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $L^2(M)$, which was recently proven by the authors in arXiv:1803.03634. We also show a characterization of Triebel-Lizorkin $\mathbf F_{p,q}^s(M)$ and Besov $\mathbf B_{p,q}^s(M)$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on manifolds $M$ with bounded geometry.