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Wavelets and self-affine tilings. (English) Zbl 0813.42021

The author generalizes the standard construction of a wavelet basis and multiresolution analyses to the case of periodic (a lattice subgroup Γ n translates the tiles transitively), self- affine (there exists a strictly expansive matrix M which maps Γ into itself and maps a tile T onto a union of m other tiles, where m=|detM|) tilings of n . The standard wavelet bases correspond to the case when Γ= n , M=2I (I=identity matrix) and m=2 n . The main result is that, assuming the existence of a self-affine lattice tiling such that the m translates of a given tile T by distinguished elements in each coset of MΓ in Γ exactly cover the affine dilation MT of T, there exists a so-called r-regular multiresolution analysis and an associated wavelet basis for every r. Here the term “r-regular” means that the functions ψ 1 ,,ψ m forming a wavelet basis associated to a multiresolution analysis satisfy the condition that they are C r and rapidly decreasing in the sense that

|(/x) α ψ i (x)|c m (1+|x|) -m

for all m and i and |α|r. A final section concerns the existence of self-similar tilings (the matrix M is a similitude).

Reviewer: J.S.Joel (Kelly)

MSC:
42C40Wavelets and other special systems
51M20Polyhedra and polytopes; regular figures, division of spaces
52C22Tilings in n dimensions (discrete geometry)
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