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The construction of \(r\)-regular wavelets for arbitrary dilations. (English) Zbl 1003.42021

The construction of compactly supported regular wavelets due to Daubechies in the case of 1 dimension is well known. However, in the case of higher dimensions, only for very specific dilation matrices such wavelets could be constructed. R. S. Strichartz [Constructive Approximation 9, No. 2-3, 327-346 (1993; Zbl 0813.42021)] proved that if a dilation matrix admits self-affine tiling, then there exists an \(r\)-regular multiresolution analysis and an associated wavelet basis. But only in the case of dimension \(\leq\) 3, every dilation matrix possesses such a property. In fact, a class of dilation matrices without such a property has been found in the work of J. C. Lagarias and Y. Wang [J. Number Theory 76, No. 2, 330-336 (1999; Zbl 1007.11066)]. Without the assumption of this “self-affine tiling property”, in this article, for any \(r\), \(r\)-regular multiresolution analysis with \(r\)-regular wavelet basis is constructed. Further, it is shown that such wavelets must necessarily have vanishing moments depending on \(r\) and the spectral properties of the dilation matrix. The article is very well written and the author has included all possible references to enable the reader to understand the problem completely.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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