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A '''multiresolution analysis''' ('''MRA''') or '''multiscale approximation''' ('''MSA''') is the design method of most of the practically relevant [[discrete wavelet transform]]s (DWT) and the justification for the [[algorithm]] of the [[fast wavelet transform]] (FWT). It was introduced in this context in 1988/89 by [[Stephane Mallat]] and [[Yves Meyer]] and has predecessors in the [[microlocal analysis]] in the theory of [[differential equation|differential equations]] (the ''ironing method'') and the [[pyramid (image processing)|pyramid method]]s of [[image processing]] as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and [http://www-prima.inrialpes.fr/Prima/Homepages/jlc/jlc.html James L. Crowley].
==
A
::<math>\{0\}\dots\subset V_1\subset V_0\subset V_{-1}\subset\dots\subset V_{-n}\subset V_{-(n+1)}\subset\dots\subset L^2(\R)</math>
that satisfies certain [[self-similarity]] relations in time
* ''Self-similarity'' in ''time'' demands that each subspace ''V<sub>k</sub>'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2<sup>k</sup>''. That is, for each <math>f\in V_k,\; m\in\Z</math> the function ''g'' defined as <math>g(x)=f(x-m2^{k})</math> also contained in <math>V_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]] factor 2<sup>''k-l''</sup>. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\R:\;g(x)=f(2^{k-l}x)</math>.
* In the sequence of subspaces, for ''k''>''l'' the space resolution 2<sup>''l''</sup> of the ''l''-th subspace is higher than the resolution 2<sup>''k''</sup> of the ''k''-th subspace.
* ''Regularity'' demands that the model [[linear subspace|subspace]] ''V<sub>0</sub>'' 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 '''[[Wavelet#Scaling function|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^2(\R) </math>, and that they are not too redundant, i.e., their [[intersection]] should only contain the [[zero element]].
== Important conclusions ==
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