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Discrete multi-resolution analysis and generalized wavelets. (English) Zbl 0777.65004
A finite number of values \(\{f^ 0_ j\}^{N_ 0}_{j=1}\) with \(N_ 0=2^{n_ 0}\) are given from which a function \(f\) is to be reconstructed (approximately). To be more precise, there may also be “coarser” sequences of \(N_ k=2^ k\) values, \(k=0,1,\dots,n_ 0-1\). The sequences constitute a discrete multi-resolution analysis if the values at the level \(k-1\), i.e. at the coarser level are determined by those on the level \(k\).
If linearity of the relation is assumed, then the weight function \(\phi\) has to satisfy a dilation equation \(\phi(x)=2\sum_ l \alpha_ l\phi(2x-l)\). This leads to a representation of \(f\) in a multi-resolution basis which is the union of generalized wavelets for all levels.
Reviewer: D.Braess (Bochum)

MSC:
65D15 Algorithms for approximation of functions
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References:
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