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| Title of object: |
wavelet set |
| Canonical Name: |
WaveletSet2 |
| Type: |
Definition |
| Created on: |
2004-06-27 13:51:48 |
| Modified on: |
2007-05-10 23:27:03 |
| Classification: |
msc:46C99 |
Preamble:
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Content:
\PMlinkescapeword{open}
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\paragraph{Definition}
An \emph{(orthonormal dyadic) wavelet set} on ${\mathbb R}$ is a subset $E \subset {\mathbb R}$ such that
\begin{enumerate}
\item $\chi_E \in L^2({\mathbb R})$ (since $\|\chi_E\| = \sqrt{m(E)}$, this implies $m(E) < \infty$).
\item $\frac{\chi_E}{\sqrt{m(E)}}$ is the Fourier transform of an orthonormal dyadic wavelet,
\end{enumerate}
where $\chi_E$ is the characteristic function of $E$, and $m(E)$ is the Lebesgue measure of $E$.
\paragraph{Characterization}
$E \subset {\mathbb R}$ is a wavelet set iff
\begin{enumerate}
\item $\{E + 2\pi n\}_{n\in {\mathbb Z}}$ is a measurable partition of $\mathbb R$; i.e. ${\mathbb R}\backslash \bigcup_{n\in \mathbb Z} \{ E + 2\pi n\}$ has measure zero, and $\bigcap_{n=i,j} \{E+2\pi n\}$ has measure zero if $i\neq j$. In short, $E$ is a $2\pi$-translation ``tiler'' of $\mathbb R$
\item $\{2^n E\}_{n\in \mathbb Z}$ is a $2$-dilation ``tiler'' of $\mathbb R$ (once again modulo sets of measure zero).
\end{enumerate}
\paragraph{Notes}
There are higher dimensional analogues to wavelet sets in $\mathbb R$, corresponding to wavelets in higher dimensions. Wavelet sets can be used to derive wavelets--- by creating a set $E$ satisfying the conditions given above, and using the inverse Fourier transform on $\chi_E$, you are guaranteed to recover a wavelet. A particularly interesting open question is: do all wavelets contain wavelet sets in their frequency support? |
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