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

Tom 118 / 2010

Eugenio Hernández, Hrvoje Šikić, Guido Weiss, Edward Wilson Colloquium Mathematicum 118 (2010), 313-332 MSC: 42C40, 43A65, 43A70. DOI: 10.4064/cm118-1-17

Streszczenie

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.

Autorzy

  • Eugenio HernándezDepartamento de Matemáticas
    Universidad Autónoma de Madrid
    28049 Madrid, Spain
    e-mail
  • Hrvoje ŠikićDepartment of Mathematics
    University of Zagreb
    Bijenička 30
    HR-10 000 Zagreb, Croatia
    e-mail
  • Guido WeissDepartment of Mathematics
    Washington University
    Box 1146
    St. Louis, MO 63130, U.S.A.
    e-mail
  • Edward WilsonDepartment of Mathematics
    Washington University
    Box 1146
    St. Louis, MO 63130, U.S.A.
    e-mail

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