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The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames

Published online by Cambridge University Press:  20 November 2018

B. Currey
Affiliation:
Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MO 63103, USA e-mail: curreybn@slu.edu
A. Mayeli
Affiliation:
Mathematics Department, Queensborough College, City University of New York, 222-05 56th Avenue Bayside, NY 11364, USA e-mail: amayeli@qcc.cuny.edu
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Abstract.

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In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi \left( \Gamma \right)\psi $, where $\pi $ is a unitary representation of a wavelet group and $\Gamma $ is the abstract pseudo-lattice $\Gamma $. We prove a sufficent condition in order that a Parseval frame $\pi \left( \Gamma \right)\psi $ can be dilated to an orthonormal basis of the form $\tau \left( \Gamma \right)\Psi $, where $\tau $ is a super-representation of $\pi $. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2013

References

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