# wavelet set

## Definition

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

1. 1.

$\chi_{E}\in L^{2}({\mathbb{R}})$ (since $\|\chi_{E}\|=\sqrt{m(E)}$, this implies $m(E)<\infty$).

2. 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. 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 $i\neq j$. In short, $E$ is a $2\pi$-translation “tiler” of $\mathbb{R}$

2. 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 WaveletSet 2013-03-22 14:27:10 2013-03-22 14:27:10 swiftset (1337) swiftset (1337) 7 swiftset (1337) Definition msc 46C99 msc 65T60 wavelet Wavelet