Multiresolution analysis: Difference between revisions

Revision as of 21:32, 22 November 2006

Revision as of 19:45, 28 November 2006

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:<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 itsthe mother wavelets integer shifts, is the orthogonal complement to <math>W_0V_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>,