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## ‣ Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

62.28425%

#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#42C40, 42B20, 58J40, 58J35, 35P05

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let
$\Delta$ 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 $f(t^2 \Delta)$. Suppose $f$ satisfies Daubechies' criterion, and
$b > 0$. For each $j$, write ${\bf M}$ as a disjoint union of measurable sets
$E_{j,k}$ with diameter at most $ba^j$, and comparable to $ba^j$ if $ba^j$ is
sufficiently small. Take $x_{j,k} \in E_{j,k}$. We then show that the functions
$\phi_{j,k}(x)=[\mu(E_{j,k})]^{1/2} \bar{K_{a^j}}(x_{j,k},x)$ form a frame for
$(I-P)L^2({\bf M})$, 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 $\bf M$ is the torus or the sphere, and $f(s)=se^{-s}$
(the "Mexican hat" situation)...

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## ‣ Continuous Wavelets on Compact Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/11/2008
Português

Relevância na Pesquisa

62.28425%

#Mathematics - Functional Analysis#Mathematics - Classical Analysis and ODEs#Mathematics - Spectral Theory#42C40, 42B20, 58J40, 58J35, 35P05

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let
$\Delta_{\bf M}$ 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 $f(t^2 \Delta_{\bf M})$. 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 $f(t^2\Delta)$ 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.

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