WaveletsMathworldPlanetmath can be used to analyze functionsMathworldPlanetmath in L2() (the space of all Lebesgue absolutely square integrable functions defined on the real numbers to the complex numbersPlanetmathPlanetmath) in much the same way the complex exponentialsMathworldPlanetmathPlanetmath are used in the Fourier transformDlmfMathworldPlanetmath, but wavelets offer the advantage of not only describing the frequency content of a function, but also providing information on the time localization of that frequency content.


A (more properly, an orthonormal dyadic) wavelet is a function ψ(t)L2() such that the family of functions


where j,k, is an orthonormal basis in the Hilbert spaceMathworldPlanetmath L2().


The scaling factor of 2j/2 ensures that ψjk=ψ=1. These type of wavelets (the most popular), are known as dyadic wavelets because the scaling factor is a power of 2. It is not obvious from the definition that wavelets even exist, or how to construct one; the Haar wavelet is the standard example of a wavelet, and one technique used to construct wavelets. Generally, wavelets are constructed from a multiresolution analysis, but they can also be generated using wavelet sets.

Title wavelet
Canonical name Wavelet
Date of creation 2013-03-22 14:26:41
Last modified on 2013-03-22 14:26:41
Owner swiftset (1337)
Last modified by swiftset (1337)
Numerical id 11
Author swiftset (1337)
Entry type Definition
Classification msc 65T60
Classification msc 46C99
Related topic FourierTransform
Related topic MultiresolutionAnalysis
Related topic WaveletSet2
Defines wavelet
Defines orthonormal dyadic wavelet