Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform

Eugenio Hernández, et al. "Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform." Colloquium Mathematicae 118.1 (2010): 313-332. <http://eudml.org/doc/286268>.

@article{EugenioHernández2010,
abstract = {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.},
author = {Eugenio Hernández, Hrvoje Šikić, Guido Weiss, Edward Wilson},
journal = {Colloquium Mathematicae},
keywords = {cyclic representation; regular representation; shift-invariant spaces; bracket functions; Zak transform},
language = {eng},
number = {1},
pages = {313-332},
title = {Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform},
url = {http://eudml.org/doc/286268},
volume = {118},
year = {2010},
}

TY - JOUR
AU - Eugenio Hernández
AU - Hrvoje Šikić
AU - Guido Weiss
AU - Edward Wilson
TI - Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform
JO - Colloquium Mathematicae
PY - 2010
VL - 118
IS - 1
SP - 313
EP - 332
AB - 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.
LA - eng
KW - cyclic representation; regular representation; shift-invariant spaces; bracket functions; Zak transform
UR - http://eudml.org/doc/286268
ER -