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Geometric aspects of frame representations of abelian groups. (English)
Trans. Am. Math. Soc. 356, No. 12, 4767-4786 (2004).
A frame for a separable Hilbert space $(H, \langle \cdot, \cdot \rangle)$ is a countable sequence $X = \{ x_j : j \in J\}$ such that there exist positive constants $C_1, C_2$ such that $C_1 \Vert x\Vert ^2 \leq \sum_{j \in J} \vert \langle x, x_j \rangle\vert ^2 \leq C_2 \Vert x\Vert ^2$ for all $x\in H$. The analysis operator $Θ$ for $X$ is given by $Θ: H \to l^2(J)$, $x\mapsto \{ \langle x, x_j \rangle: j \in J \}$. Let $G$ be a countable abelian group and $π: G \to B(H)$ a unitary representation of $G$ on a Hilbert space $H$. Such a representation is called a frame representation if there is a frame vector $v\in H$ such that $\{ π(g) v : g \in G\}$ is a frame for $H$. To each frame representation a multiplicity function $m : \widehat{G} \to \{ 0, 1, \ldots, \infty \}$ is associated, where $\widehat{G}$ denotes the character group of $G$. Let $π_H$ and $π_K$ be two frame representations of $G$ on $H$ and $K$ with analysis operators $Θ_H$ and $Θ_K$ for the corresponding frame vectors, respectively. Then a characterisation for the ranges of $Θ_H$ and $Θ_K$ to be equal, to be orthogonal, to have non-trivial intersection, or for one being contained in the other is obtained, where the characterisation is in terms of the supports of the corresponding multiplicity functions. Extensions to Bessel sequences arising from the action of the group are also considered. The results are then applied to the sampling of bandlimited functions and to wavelet and Weyl-Heisenberg frames, giving sufficient conditions for the above mentioned properties of the ranges of the analysis operators. The multiplicity functions are obtained explicitly in these cases.
Alexander Lindner (München)