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