Continuous wavelets on compact manifolds

By Daryl Geller, Azita Mayeli. 2008

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Let M be a smooth compact oriented Riemannian manifold, and let ΔM 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 Kt(x, y) denote the kernel of f (t2 ΔM). 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 (t2Δ) on $${\mathbb {R}^n}$$ . We define continuous $${\mathcal {S}}$$-wavelets on M, in such a manner that Kt(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 S2, 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/6939d06ed9c8ee137232a2ecea7478ff44bad06c