Nearly tight frames and space-frequency analysis on compact manifolds |
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Authors: | Daryl Geller Azita Mayeli |
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Affiliation: | (1) Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA |
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Abstract: | Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say , and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 Δ). Suppose f satisfies Daubechies’ criterion, and b > 0. For each j, write M as a disjoint union of measurable sets E j,k with diameter at most ba j , and measure comparable to if ba j is sufficiently small. Take x j,k ∈ E j,k . We then show that the functions form a frame for (I − P)L 2(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I − P)L 2 is in space and in frequency, we can describe which terms in the summation are so small that they can be neglected. If n = 2 and M is the torus or the sphere, and f (s) = se −s (the “Mexican hat” situation), we obtain two explicit approximate formulas for the φ j,k , 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. |
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Keywords: | Frames Wavelets Continuous wavelets Spectral theory Schwartz functions Time– frequency analysis Manifolds Sphere Torus Pseudodifferential operators |
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