Continuous wavelets on compact manifolds

Abstract

Let M be a smooth compact oriented Riemannian manifold, and let M 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 M ). We show that K t 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 (t 2 ) on R n . We define continuous S-wavelets on M, in such a manner that K t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on R n in S(R n ). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus T 2 or the sphere S 2 , and f (s) = se −s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K t , one to be used when t is large, and one to be used when t is small.

Besov spaces and frames on compact manifolds

“…We shall need the following basic facts, from section 3 of [7], about M and its geodesic distance d. For x ∈ M, we let B(x, r) denote the ball {y : d(x, y) < r}. …”

“…For ν ≥ −1, let J ν be the kernel of β ν (∆). Using the eigenfunction expansion of β ν (∆) (see (25) of [7]), one sees at once that J −1 (x, y) is smooth in (x, y). Moreover, for ν ≥ 0, β ν ∈ S(R + ) and β ν (0) = 0; so J ν is smooth as well.…”

“…Proof By the last sentence of Lemma 1.2, if we let K 0 t (x, y) be the kernel of f 0 (t 2 ∆), and if we let η j,k (y) = K 0 a j (x j,k , y) for j ≤ 0, then for every I, J there exists C IJ with η j,k ∈ C IJ M IJ x j,k ,a j . Also (for instance, by looking at eigenfunction expansions, as in (25) of [7]), one has that ϕ j,k = (a 2j ∆) l η j,k . Thus, by Theorem 3.5, if F ∈ B αq p , the series (*) There exist N, C 0 (independent of our choices of b, E j,k , x j,k ) as follows.…”

“…To justfiy the formal argument leading from (7) to (12), and to go beyond the L 2 theory, one needs the following information about the kernel K t , which we established in Lemma 4.1 of our earlier paper [7] (see also the remark following the proof of that lemma): Lemma 1.2 Say f (0) = 0. Then for every pair of C ∞ differential operators X (in x) and Y (in y) on M, and for every integer N ≥ 0, there exists C N,X,Y as follows.…”

The Data Big Bang and the Expanding Digital Universe: High‐Dimensional, Complex and Massive Data Sets in an Inflationary Epoch

“…Geller et al [113] adopted the spin needlet approach for the analysis of CMB polarization measurements. Lately Geller and Mayeli [114] constructed continuous wavelets and nearly tight frames (needlets) on compact manifolds. The essential part of the general analysis based on the wavelet-like constructions is sampling of bandlimited functions.…”