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

1.
${\chi}_{E}\in {L}^{2}(\mathbb{R})$ (since $\parallel {\chi}_{E}\parallel =\sqrt{m(E)}$, this implies $$).

2.
$\frac{{\chi}_{E}}{\sqrt{m(E)}}$ is the Fourier transform of an orthonormal dyadic wavelet,
where ${\chi}_{E}$ is the characteristic function^{} of $E$, and $m(E)$ is the Lebesgue measure^{} of $E$.
Characterization
$E\subset \mathbb{R}$ is a wavelet set iff

1.
${\{E+2\pi n\}}_{n\in \mathbb{Z}}$ 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 $i\ne j$. In short, $E$ is a $2\pi $translation “tiler” of $\mathbb{R}$

2.
${\{{2}^{n}E\}}_{n\in \mathbb{Z}}$ is a $2$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 $E$ 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^{}?
Title  wavelet set 

Canonical name  WaveletSet 
Date of creation  20130322 14:27:10 
Last modified on  20130322 14:27:10 
Owner  swiftset (1337) 
Last modified by  swiftset (1337) 
Numerical id  7 
Author  swiftset (1337) 
Entry type  Definition 
Classification  msc 46C99 
Classification  msc 65T60 
Related topic  wavelet 
Related topic  Wavelet 