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# The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames

Published:2013-03-25
Printed: Dec 2013
• B. Currey,
Department of Mathematics and Computer Science, Saint Louis University, St. Louis, MO 63103, USA
• A. Mayeli,
Mathematics Department, Queensborough College, City University of New York, 222-05 56th Avenue Bayside, NY 11364, USA
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## Abstract

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 $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ 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.
 Keywords: frame, dilation, wavelet, Baumslag-Solitar group, shearlet
 MSC Classifications: 43A65 - Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45] 42C40 - Wavelets and other special systems 42C15 - General harmonic expansions, frames