Multiresolution analysis: Difference between revisions

Revision as of 18:23, 8 December 2005

Revision as of 18:34, 8 December 2005

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== Important conclusions ==

This is only forIn the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, (theone may make a number of deductions. The proof of existence of thosethis class of functions is due to [[Ingrid Daubechies]]).

Then thereThere is, because of <math>V_0\subset V_1</math>, a finite sequence of coefficients <math>a_k=2< \langle\phi(x),\phi(2x-k)>\rangle</math>, for <math>|k|\leq N</math> and <math>a_k=0</math> for <math>|k|>N</math>, such that

This is only for the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts (the existence of those is due to [[Ingrid Daubechies]]).

Then there is, because of <math>V_0\subset V_1</math>, a finite sequence of coefficients <math>a_k=2<\phi(x),\phi(2x-k)></math>, <math>a_k=0</math> for |k|>N, such that

:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k)</math>.

Defining another function, known as '''mother wavelet''' or just '''the wavelet'''

:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k)</math>,

one can see that the space <math>W_0\subset V_1</math>, which is defined as the linear hull of its integer shifts, is the orthogonal complement to <math>W_0</math> inside <math>V_1</math>. Or put differently, <math>V_1</math> is the orthogonal sum of <math>W_0</math> and <math>V_0</math>. By self-similarity, there are scaled versions <math>W_k</math> of <math>W_0</math> and by completeness one has

:<math>L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k</math>,

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:<math>\{\psi_{k,n}(x)=\sqrt2^k\psi(2^kx-n):\;k,n\in\Z\}</math>

is a countable complete [[orthonormal systemwavelet]] basis in <math>L^2(\R)</math>.

==References==