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

Eugenio Hernández ¹   ; Hrvoje Šikić ²   ; Guido Weiss ³   ; Edward Wilson ³

¹ Departamento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain
² Department of Mathematics University of Zagreb Bijenička 30 HR-10 000 Zagreb, Croatia
³ Department of Mathematics Washington University Box 1146 St. Louis, MO 63130, U.S.A.

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 313-332

Résumé

We just published a paper showing that the properties of the shift invariant spaces, $\langle f\rangle$, generated by the translates by $\mathbb{Z}^n$ of an $f$ in $L^2(\mathbb{R}^n)$ correspond to the properties of the spaces $L^2(\mathbb{T}^n,p)$, where the weight $p$ equals $[\hat f,\hat f]$. This correspondence helps us produce many new properties of the spaces $\langle f\rangle$. In this paper we extend this method to the case where the role of $\mathbb{Z}^n$ is taken over by locally compact abelian groups $G$, $L^2(\mathbb{R}^n)$ is replaced by a separable Hilbert space on which a unitary representation of $G$ acts, and the role of $L^2(\mathbb{T}^n,p)$ is assumed by a weighted space $L^2(\widehat G, w)$, where $\widehat G$ is the dual group of $G$. This provides many different extensions of the theory of wavelets and related methods for carrying out signal analysis.

Détail

DOI : 10.4064/cm118-1-17

Keywords: just published paper showing properties shift invariant spaces langle rangle generated translates mathbb mathbb correspond properties spaces mathbb where weight equals hat hat correspondence helps produce many properties spaces langle rangle paper extend method where role mathbb taken locally compact abelian groups mathbb replaced separable hilbert space which unitary representation acts role mathbb assumed weighted space widehat where widehat dual group nbsp provides many different extensions theory wavelets related methods carrying out signal analysis