wavelet set

Definition

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

1. ${\chi }_E\in L^2(ℝ)$ (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

$E\subset ℝ$ is a wavelet set iff

1. ${\{E+2\pi n\}}_{n\in \mathbb{Z}}$ is a measurable partition of $ℝ$; i.e. $ℝ\backslash \backslash {\bigcup }_{n\in \mathbb{Z}}\{E+2\pi n\}$ 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 $ℝ$

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

Notes

There are higher dimensional analogues to wavelet sets in $ℝ$, 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?