Multiresolution analysis: Difference between revisions

Revision as of 14:11, 30 December 2020

Revision as of 14:58, 10 January 2021

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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 show that the space <math>W_0\subset V_{-1}</math>, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to <math>V_0</math> inside <math>V_{-1}</math>.<ref>{{Cite web|url=http://www.cmap.polytechnique.fr/~mallat/book.html|title=A Wavelet Tour of Signal Processing|last=Mallat, S.G.|first=|date=|website=www.di.ens.fr|url-status=live|archive-url=|archive-date=|access-date=2019-12-30}}</ref> Or put differently, <math>V_{-1}</math> is the [[orthogonal direct sum|orthogonal sum]] (denoted by <math>\oplus</math>) 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(\R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>

thus the set