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Frames and Single Wavelets for Unitary Groups

Published:2002-06-01
Printed: Jun 2002
• Eric Weber
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Abstract

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$.
 Keywords: wavelet, multiresolution analysis, unitary group representation, frame
 MSC Classifications: 42C40 - Wavelets and other special systems 43A25 - Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 42C15 - General harmonic expansions, frames 46N99 - None of the above, but in this section