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Changed sign convention so that V_k has space resolution 2^k, compatible with the continuous transforms in the wavelet article. |
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::<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/space and scale/frequency, as well as [[Complete metric space|completeness]] and regularity relations.
* ''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>.
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