"Multiplication Generated Duals of a Subspace Frame and its Applications"
Sarkar, SudiptaDue to the redundancy property of frames, the stable decomposition of a vector in the separable Hilbert space $\mathcal H$ allows the flexibility of choosing different types of duals for a frame. In this article, we study the properties of alternate (oblique) $\Gamma$-translation generated (TG) duals for a continuous frame in $L^2(\mathscr G)$, where $\mathscr G$ is a locally compact group (not necessarily abelian) and $\Gamma$ is a closed abelian subgroup. First, we establish the theory of such decompositions for the multiplication generated system on the Bochner space $L^2(X; \mathcal H)$ using the range function corresponding to the point-wise conditions in $\mathcal H$, where $X$ is a $\sigma$-finite measure space. Then, we obtain characterizations of these $\Gamma$-TG dual systems (need not be a frame) in terms of Zak transform for the pair $(\mathscr G, \Gamma)$ and also analyze the reduction of the continuous problem in $L^2 (X; \mathcal H)$ to a discrete problem in $\mathcal H$ for countable functions. Further, we characterize these duals' uniqueness using Gramian/dual Gramian operators, which become a discrete frame/Riesz basis for the associated range space.
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