Multiresolution analysis: Difference between revisions

Revision as of 12:55, 26 February 2013

Revision as of 02:00, 12 March 2013

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In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to [[Ingrid Daubechies]].

Assuming the scaling function has compact support, then <math>V_0\subset V_1</math> implies that there is a finite sequence of coefficients <math>a_k=2 \langle\phi(x),\phi(2x-k)\rangle</math>, for <math>|k|\leq N</math>, and <math>a_k=0</math> for <math>|k|>N</math>, such that

:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k).</math>