Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform
Tom 118 / 2010
Streszczenie
We just published a paper showing that the properties of the shift invariant spaces, , 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.
