Continuous Wavelets on Compact Manifolds

Let M be a smooth compact oriented Riemannian manifold, and let ∆M be the Laplace-Beltrami operator on M. Say 0 6= f ∈ S(R), and that f(0) = 0. For t > 0, let Kt(x, y) denote the kernel of f(t∆M). We show that Kt 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∆) on R. We define continuous S-wavelets on M, in such a manner that Kt(x, y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on R in S(R). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S−wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus T or the sphere S, and f(s) = se−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for Kt, one to be used when t is large, and one to be used when t is small.