The unitary extension principle on locally compact abelian groups

Ole Christensen*, Say Song Goh

*Corresponding author for this work

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Abstract

The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for L2(Rs) based on a given refinable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions; via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.
Original languageEnglish
JournalApplied and Computational Harmonic Analysis
Number of pages33
ISSN1063-5203
DOIs
Publication statusPublished - 2017

Keywords

  • Frames
  • The unitary extension principle
  • LCA group

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