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Abstract: A traditional wavelet is a special case of a vector in a separable Hilbertspace that generates a basis under the action of a system of unitary operatorsdefined in terms of translation and dilation operations. ACoxeter-fractal-surface wavelet is obtained by defining fractal surfaces onfoldable figures, which tesselate the embedding space by reflections in theirbounding hyperplanes instead of by translations along a lattice. Although boththeories look different at their onset, there exist connections andcommunalities which are exhibited in this semi-expository paper. In particular,there is a natural notion of a dilation-reflection wavelet set. We prove thatdilation-reflection wavelet sets exist for arbitrary expansive matrixdilations, paralleling the traditional dilation-translation wavelet theory.There are certain measurable sets which can serve simultaneously asdilation-translation wavelet sets and dilation-reflection wavelet sets,although the orthonormal structures generated in the two theories areconsiderably different.



Author: David Larson, Peter Massopust

Source: https://arxiv.org/







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