The unitary extension principle for locally compact abelian groups with co-compact subgroups

Christensen, Ole; Goh, Say Song

doi:10.1090/proc/15319

The unitary extension principle for locally compact abelian groups with co-compact subgroups
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by Ole Christensen and Say Song Goh

Proc. Amer. Math. Soc. 149 (2021), 1189-1202

DOI: https://doi.org/10.1090/proc/15319

Published electronically: January 22, 2021

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Abstract:

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for $L^{2}(\mathbb {R})$ and associated expansions of functions $f\in L^{2}(\mathbb {R})$ of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

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