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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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