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单词 Wavelet
释义

wavelet


Motivation

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.

Definition

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

ψjk2j/2ψ(2jt-k),

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

Notes

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.

Titlewavelet
Canonical nameWavelet
Date of creation2013-03-22 14:26:41
Last modified on2013-03-22 14:26:41
Ownerswiftset (1337)
Last modified byswiftset (1337)
Numerical id11
Authorswiftset (1337)
Entry typeDefinition
Classificationmsc 65T60
Classificationmsc 46C99
Related topicFourierTransform
Related topicMultiresolutionAnalysis
Related topicWaveletSet2
Defineswavelet
Definesorthonormal dyadic wavelet
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更新时间:2025/5/4 6:40:43