Nearly tight frames and space-frequency analysis on compact manifolds

By Daryl Geller, Azita Mayeli. 2007

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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 $${0 \neq f \in \mathcal{S}(\mathbb R^+)}$$ , and that f (0)  =  0. For t  >  0, let Kt(x, y) denote the kernel of f (t2 Δ). Suppose f satisfies Daubechies’ criterion, and b  >  0. For each j, write M as a disjoint union of measurable sets Ej,k with diameter at most baj, and measure comparable to $${(ba^j)^n}$$ if baj is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions $${\phi_{j,k}(x)=\mu(E_{j,k})^{1/2} \overline{K_{a^j}}(x_{j,k},x)}$$ form a frame for (I  −  P)L2(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)L2 is in space and in frequency, we can describe which terms in the summation $${F \sim SF = \sum_j \sum_k \langle F,\phi_{j,k} \rangle \phi_{j,k}}$$ 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 φj,k, one to be used when t is large, and one to be used when t is small.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/823fa8dc8b74230d3fa48d12a5d98ad3c25642e6

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In Applied Mathematics

  1. Daryl Geller
  2. Azita Mayeli

In Computational Theory and Mathematics

  1. Daryl Geller
  2. Azita Mayeli

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