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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 $\Gamma\subset \bbfR\sp n$ translates the tiles transitively), self- affine (there exists a strictly expansive matrix $M$ which maps $\Gamma$ into itself and maps a tile $T$ onto a union of $m$ other tiles, where $m= \vert\text{det }M\vert)$ tilings of $\bbfR\sp n$. The standard wavelet bases correspond to the case when $\Gamma= \bbfZ\sp n$, $M= 2I$ ($I$=identity matrix) and $m= 2\sp n$. The main result is that, assuming the existence of a self-affine {\it lattice} tiling such that the $m$ translates of a given tile $T$ by distinguished elements in each coset of $M\Gamma$ in $\Gamma$ 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 $\psi\sb 1,\dots, \psi\sb m$ forming a wavelet basis associated to a multiresolution analysis satisfy the condition that they are $C\sp r$ and rapidly decreasing in the sense that $$\vert(\partial/\partial x)\sp \alpha \psi\sb i(x)\vert \le c\sb m(1+ \vert x\vert)\sp{-m}$$ for all $m$ and $i$ and $\vert\alpha\vert\le r$. A final section concerns the existence of self-similar tilings (the matrix $M$ is a similitude).
Reviewer: J.S.Joel (Kelly)

##### MSC:
 42C40 Wavelets and other special systems 51M20 Polyhedra and polytopes; regular figures, division of spaces 52C22 Tilings in $n$ dimensions (discrete geometry)
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##### References:
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