Dai, Xingde.Larson, David R.Speegle, Darrin m.2019-05-082019-05-082015-06-18https://doi.org/10.1007/BF02649106http://erepository.kibu.ac.ke/handle/123456789/900A 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 (ℝ n). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ n.enAttribution-NonCommercial-ShareAlike 3.0 United Stateshttp://creativecommons.org/licenses/by-nc-sa/3.0/us/Measurable subsetNonempty interiorOrthogonal waveletOrthonormal waveletMeasurable partitionArticle