PlanetMath: wavelet set

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wavelet set

(Definition)

Definition

An (orthonormal dyadic) wavelet set on ${\mathbb{R}}$ is a subset $E \subset {\mathbb{R}}$ such that

$\chi_E \in L^2({\mathbb{R}})$ (since $\Vert\chi_E\Vert = \sqrt{m(E)}$ , this implies $m(E) < \infty$ ).
$\frac{\chi_E}{\sqrt{m(E)}}$ is the Fourier transform of an orthonormal dyadic wavelet,

where $\chi_E$ is the characteristic function of

, and

is the Lebesgue measure of

Characterization

$E \subset {\mathbb{R}}$ is a wavelet set iff

$\{E + 2\pi n\}_{n\in {\mathbb{Z}}}$ 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, is a $2\pi$ -translation “tiler” of $\mathbb{R}$
$\{2^n E\}_{n\in \mathbb{Z}}$ is a -dilation “tiler” of $\mathbb{R}$ (once again modulo sets of measure zero).

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

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?

"wavelet set" is owned by swiftset.

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See Also: wavelet, wavelet

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Cross-references: support, open question, wavelets, measure zero, measurable partition, iff, Lebesgue measure, characteristic function, orthonormal dyadic wavelet, Fourier transform, implies, subset, dyadic, orthonormal
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This is version 4 of wavelet set, born on 2004-06-27, modified 2007-09-27.
Object id is 5971, canonical name is WaveletSet2.
Accessed 2057 times total.

Classification:

AMS MSC:	46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous)
	65T60 (Numerical analysis :: Numerical methods in Fourier analysis :: Wavelets)

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