, Volume 262, Issue 4, pp 895-927

Continuous wavelets on compact manifolds

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Abstract

Let M be a smooth compact oriented Riemannian manifold, and let Δ M be the Laplace–Beltrami operator on M. Say ${0 \neq f \in \mathcal{S}(\mathbb {R}^+)}$ , and that f (0)  =  0. For t  >  0, let K t (x, y) denote the kernel of f (t 2 Δ 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 2Δ) on ${\mathbb {R}^n}$ . We define continuous ${\mathcal {S}}$ -wavelets on 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 M are analogous to continuous wavelets on ${\mathbb {R}^n}$ in ${\mathcal {S}}$ ( ${\mathbb {R}^n}$ ). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous ${\mathcal {S}}$ -wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus ${\mathbb T^2}$ 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.

A. Mayeli was partially supported by the Marie Curie Excellence Team Grant MEXT-CT-2004-013477, Acronym MAMEBIA.