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Journal of Geometric Analysis

, Volume 19, Issue 1, pp 19–35 | Cite as

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

  • Jeffrey D. BlanchardEmail author
Article

Abstract

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L 2(ℝ n ) under the actions of translations from a full rank lattice and dilations by products of elements of groups A and B. A minimally supported frequency composite dilation Parseval frame wavelet has generating functions whose Fourier transforms are characteristic functions of sets contained in a lattice tiling set. Constructive proofs are used to establish the existence of minimally supported frequency composite dilation Parseval frame wavelets in arbitrary dimension using any finite group B, any full rank lattice, and an expanding matrix generating the group A and normalizing the group B. Moreover, every such system is derived from a Parseval frame multiresolution analysis. Multiple examples are provided including examples that capture directional information.

Keywords

Affine systems Composite dilation wavelets Coxeter groups Minimally supported frequency Parseval frames Wavelet frames Wavelet sets Wavelets 

Mathematics Subject Classification (2000)

42C15 42C40 20F55 51F15 

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Copyright information

© Mathematica Josephina, Inc. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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