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On the duality of fusion frames. (English) Zbl 1127.46016
A system $(f_i)_{i\in I}$ is a frame in a Hilbert space $\Cal H$ if there are positive constants $A$ and $B$ such that $A\Vert f\Vert ^2 \leq \sum_{i\in I} |\langle f,f_i\rangle\vert ^2 \leq B\Vert f\Vert ^2$ for all $f\in \Cal H$. A system $\Cal V= ((V_i,V_i))_{i\in I}$ is a fusion frame or a frame of subspaces if $$ A\Vert f\Vert ^2 \leq \sum_{i\in I} v_i^2 \Vert \pi_{V_i}(f)\Vert^2 \leq B\Vert f\Vert ^2 $$ where $\pi_V$ is the orthogonal projection onto the subspace $V$. One of the main results of this paper is a proof that the dual fusion frame $((S_{\Cal V}^{-1}V_i, v_i))_{i\in I}$ (with $S_{\Cal V}$ the frame operator given by $\sum_{i\in I} v_i\pi_{V_i}(f)$) is indeed a fusion frame. Other results deal with alternate duals, i.e., systems $\Cal W=((W_i,w_i))_{i\in I}$ so that $f = \sum_{i\in I} v_iw_i \pi_{W_i}S_{\Cal V}^{-1}\pi_{V_i}(f)$, and frame operators for a pair of two Bessel fusion sequences (where only the upper bound above is required to hold).

MSC:
46C15Characterizations of Hilbert spaces
42C99Non-trigonometric Fourier analysis
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References:
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