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Frames and Single Wavelets for Unitary Groups
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- Journal:
- Canadian Journal of Mathematics / Volume 54 / Issue 3 / 01 June 2002
- Published online by Cambridge University Press:
- 20 November 2018, pp. 634-647
- Print publication:
- 01 June 2002
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- Article
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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 multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on
${{L}^{2}}(\mathbb{R})$.
