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 be the Laplace-Beltrami operator on M. Say 0 = f ∈ S(R + ), and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 ). 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 toform 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 ∼ S F = j k F, φ j,k φ 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.

Besov spaces and frames on compact manifolds

“…As frames they use the "needlets" that they constructed in [16]. We discussed the similarities, advantages and disadvantages of these frames as compared to ours on S n , in section 3 of our earlier article [8]. They proved and used a result similar to our Lemma 1.2, and our methods (based on adapting the ideas in [2]) are rather similar to theirs.…”

“…(In [8] we show that such E j,k exist provided c 0 and δ 0 are sufficiently small, independent of the values of a and b.) For 1 ≤ k ≤ N j , define φ j,k by (12).…”

“…Say then that 0 < p < 1. By the discussion before Theorem 2.4 of [8], for each j ≥ 0, there exists a finite covering of M by disjoint measurable sets…”

“…(We hope to consider Triebel-Lizorkin spaces in a forthcoming article.) Our methods, in addition to using the results of [8], are largely adapted from those of Frazier and Jawerth [2], who gave a similar characterization of Besov spaces on R n . However, as we shall explain below, some new ideas are needed on manifolds.…”

“…In order for the notation to be fully analogous to that in [2], we shall need to adapt the notation that we used in our earlier article [8]; through much of this article, we will write…”