Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform
Volume 118 / 2010
Abstract
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.
