Continuous Wavelets on Compact Manifolds

• Published in: arXiv.org 2008-11
• Main Authors: Geller, Daryl, Mayeli, Azita
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Convolution; Kernels; Operators (mathematics); Riemann manifold; Toruses; Wavelet transforms
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0811.4440

Let \(\bf M\) be a smooth compact oriented Riemannian manifold, and let \(\Delta_{\bf M}\) be the Laplace-Beltrami operator on \({\bf M}\). Say \(0 \neq f \in \mathcal{S}(\RR^+)\), and that \(f(0) = 0\). For \(t > 0\), let \(K_t(x,y)\) denote the kernel of \(f(t^2 \Delta_{\bf M})\). We show that \(K_t\) is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator \(f(t^2\Delta)\) on \(\RR^n\). We define continuous \({\cal S}\)-wavelets on \({\bf M}\), in such a manner that \(K_t(x,y)\) satisfies this definition, because of its localization near the diagonal. Continuous \({\cal S}\)-wavelets on \({\bf M}\) are analogous to continuous wavelets on \(\RR^n\) in \(\mathcal{S}(\RR^n)\). In particular, we are able to characterize the H\(\ddot{o}\)lder continuous functions on \({\bf M}\) by the size of their continuous \({\mathcal{S}}-\)wavelet transforms, for H\(\ddot{o}\)lder exponents strictly between 0 and 1. If \(\bf M\) is the torus \(\TT^2\) or the sphere \(S^2\), and \(f(s)=se^{-s}\) (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for \(K_t\), one to be used when \(t\) is large, and one to be used when \(t\) is small.