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Wavelet sets in \(\mathbb{R}^ n\). (English) Zbl 0881.42023
Summary: A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on \(L^2(\mathbb{R}^n)\). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure of \(\mathbb{R}^n\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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