Continuous Wavelets on Compact Manifolds
| dc.creator | Geller, Daryl | |
| dc.creator | Mayeli, Azita | |
| dc.date | 2008-11-26 | |
| dc.date.accessioned | 2026-07-07T12:06:07Z | |
| dc.date.available | 2026-07-07T12:06:07Z | |
| dc.description | Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $Δ_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 Δ_{\bf M})$. 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 $f(t^2Δ)$ on $\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus $\TT^2$ 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. | |
| dc.identifier | https://arxiv.org/abs/0811.4440 | |
| dc.identifier | http://arxiv.org/abs/0811.4440 | |
| dc.identifier | doi:10.1007/s00209-008-0405-7 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/208566 | |
| dc.subject | Functional Analysis | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Spectral Theory | |
| dc.subject | 42C40, 42B20, 58J40, 58J35, 35P05 | |
| dc.title | Continuous Wavelets on Compact Manifolds | |
| dc.type | text |