Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds

Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operator on M. Say 0 6= f ∈ S(R), and that f(0) = 0. For t > 0, let Kt(x, y) denote the kernel of f(t∆). 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 ba j , and comparable to ba if ba is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions φj,k(x) = [μ(Ej,k)]Kaj (xj,k, x) form a frame for (I − P )L(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 ∈ L is in space and in frequency, we can describe which terms in the summation F ∼ SF = P j P k〈F, φj,k〉φj,k are so small that they can be neglected. Finally we explain in what sense the kernel Kt(x, y) should itself be regarded as a continuous wavelet on M, and characterize the Hölder continuous functions on M by the size of their continuous wavelet transforms, for Hölder exponents strictly between 0 and 1.