2007/10/19 by David R. Larson, Larson, David, Peter Massopust +1
Mathematics · Physics and Astronomy · #20F55 #28A80 #42C40 #51F15 #Advanced Mathematical Theories and Applications #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.0710.3655
openalex publication_date 2007/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different.