Continuous Wavelets on Compact Manifolds

Abstract

Let $\mathbf{M}$ be a smooth compact oriented Riemannian manifold, and let ${\Delta }_{\mathbf{M}}$ be the Laplace-Beltrami operator on $\mathbf{M}$. 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 $f(t^2{\Delta }_{\mathbf{M}})$. 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 $\mathcal{S}$-wavelets on $\mathbf{M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous $\mathcal{S}$-wavelets on $\mathbf{M}$ 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 $\mathbf{M}$ by the size of their continuous $\mathcal{S}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\mathbf{M}$ is the torus $\TT^2$ or the sphere $S^2$, and $f(s)=s{e}^{-s}$ (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.

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