Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds

Let $\mathbf{M}$ be a smooth compact oriented Riemannian manifold, and let $\mathrm{\Delta }$ be the Laplace-Beltrami operator on $\mathbf{M}$. Say $0 \neq f \in $\mathcal{S}$(\RR^+)$,andthat$f(0) = 0$\mathrm{.}For$t > 0$,let$K_t(x,y)$ denote the kernel of $f(t^2\mathrm{\Delta })$. Suppose $f$ satisfies Daubechies’ criterion, and $b>0$. For each $j$, write $\mathbf{M}$ as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $b{a}^j$, and comparable to $b{a}^j$ if $b{a}^j$ is sufficiently small. Take $x_{j,k}\in E_{j,k}$. We then show that the functions ${\varphi }_{j,k}(x)=[\mu (E_{j,k}){]}^{1\mathrm{/}2}\overline{K_{a^j}}(x_{j,k},x)$ form a frame for $(I-P)L^2(\mathbf{M})$, 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 $F\in (I-P)L^2$ is in space and in frequency, we can describe which terms in the summation $F \sim SF = \sum_j \sum_k < F,\phi_{j,k} > \phi_{j,k}$ are so small that they can be neglected. If $n=2$ and $\mathbf{M}$ is the torus or the sphere, and $f(s)=s{e}^{-s}$ (the “Mexican hat” situation), we obtain two explicit approximate formulas for the ${\varphi }_{j,k}$, one to be used when $t$ is large, and one to be used when $t$ is small. Finally we explain in what sense the kernel $K_t(x,y)$ should itself be regarded as a continuous wavelet on $\mathbf{M}$, and characterize the H"older continuous functions on $\mathbf{M}$ by the size of their continuous wavelet transforms, for H"older exponents strictly between 0 and 1.