Nearly tight frames and space-frequency analysis on compact manifolds

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 $${0 \neq f \in \mathcal{S}(\mathbb R^+)}$$ , 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 $${(ba^j)^n}$$ if ba j is sufficiently small. Take x j,k ∈ E j,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)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 $${F \sim SF = \sum_j \sum_k \langle F,\phi_{j,k} \rangle \phi_{j,k}}$$ 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 φ j,k, one to be used when t is large, and one to be used when t is small.