wavelet set


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

  1. 1.

    χEL2() (since χE=m(E), this implies m(E)<).

  2. 2.

    χEm(E) is the Fourier transform of an orthonormal dyadic wavelet,

where χE is the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of E, and m(E) is the Lebesgue measureMathworldPlanetmath of E.


E is a wavelet set iff

  1. 1.

    {E+2πn}n is a measurable partition of ; i.e. \n{E+2πn} has measure zeroMathworldPlanetmath, and n=i,j{E+2πn} has measure zero if ij. In short, E is a 2π-translation “tiler” of

  2. 2.

    {2nE}n is a 2-dilation “tiler” of (once again modulo sets of measure zero).


There are higher dimensional analogues to wavelet sets in , 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 χE, you are guaranteed to recover a wavelet. A particularly interesting open question is: do all wavelets contain wavelet sets in their frequency supportMathworldPlanetmath?

Title wavelet set
Canonical name WaveletSet
Date of creation 2013-03-22 14:27:10
Last modified on 2013-03-22 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