Multiresolution analysis: Difference between revisions

Revision as of 04:36, 18 January 2007

Revision as of 23:26, 24 June 2007

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* ''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]].

* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be [[dense set|dense]] in ''<math>L&sup2;^2(IR\mathbb{R})''</math>, and that they are not too redundant, i.e., their intersection should only contain the zero element.

== Important conclusions ==