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The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames
Published online by Cambridge University Press: 20 November 2018
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 \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.
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- Copyright © Canadian Mathematical Society 2013
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