Mathematische Zeitschrift

, 262:895 | Cite as

Continuous wavelets on compact manifolds

Article

Abstract

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)  =  ses (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.

Keywords

Frames Wavelets Continuous Wavelets Spectral Theory Schwartz Functions Time-Frequency Analysis Manifolds Sphere Torus Pseudodifferential Operators Hölder Spaces 

Mathematics Subject Classification (2000)

42C40 42B20 58J40 58J35 35P05 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA

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