Multiresolution analysis: Difference between revisions

Revision as of 11:53, 17 April 2012

Revision as of 15:39, 22 April 2012

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* ''Self-similarity'' in ''time'' demands that each subspace ''V_k'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2^-k''. That is, for each <math>f\in V_k,\; m\in\mathbb Z</math> there is a <math>g\in V_k</math> with <math>\forall x\in\mathbb R:\;f(x)=g(x+m2^{-k})</math>.

* ''Self-similarity'' in ''scale'' demands that all subspaces <math>V_k\subset V_l,\; k<l,</math> are time-scaled versions of each other, with [[Scaling_(geometry)|scaling]] respectively [[Dilation (metric space)|dilation]]{{dn|date=April 2012}} factor ''2^l-k''. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\mathbb R:\;g(x)=f(2^{l-k}x)</math>. If f has limited [[support (mathematics)|support]], then as the support of g gets smaller, the resolution of the ''l''-th subspace is higher than the resolution of the ''k''-th subspace.

* ''Regularity'' demands that the model subspace ''V₀'' be generated as the [[linear hull]] ([[algebraic closure|algebraically]] or even [[topologically closed]]) of the integer shifts of one or a finite number of generating functions <math>\phi</math> or <math>\phi_1,\dots,\phi_r</math>. Those integer shifts should at least form a frame for the subspace <math>V_0\subset L^2(\R)</math>, which imposes certain conditions on the decay at infinity. The generating functions are also known as '''scaling functions''' or '''father wavelets'''. In most cases one demands of those functions to be [[piecewise continuous]] with [[compact support]].