We just published a paper showing that the properties of the shift invariant spaces, ⟨f⟩, generated by the translates by ℤⁿ of an f in L²(ℝⁿ) correspond to the properties of the spaces L²(𝕋ⁿ,p), where the weight p equals [f̂,f̂]. This correspondence helps us produce many new properties of the spaces ⟨f⟩. In this paper we extend this method to the case where the role of ℤⁿ is taken over by locally compact abelian groups G, L²(ℝⁿ) is replaced by a separable Hilbert space on which a unitary representation of G acts, and the role of L²(𝕋ⁿ,p) is assumed by a weighted space L²(Ĝ,w), where Ĝ is the dual group of G. This provides many different extensions of the theory of wavelets and related methods for carrying out signal analysis.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-17,
author = {Eugenio Hern\'andez and Hrvoje \v Siki\'c and Guido Weiss and Edward Wilson},
title = {Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform},
journal = {Colloquium Mathematicae},
volume = {120},
year = {2010},
pages = {313-332},
zbl = {1203.42048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-17}
}
Eugenio Hernández; Hrvoje Šikić; Guido Weiss; Edward Wilson. Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform. Colloquium Mathematicae, Tome 120 (2010) pp. 313-332. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-1-17/