Continuous wavelets on compact manifolds

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.