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On the duality of fusion frames. (English) Zbl 1127.46016

A system ${\left({f}_{i}\right)}_{i\in I}$ is a frame in a Hilbert space $ℋ$ if there are positive constants $A$ and $B$ such that ${A\parallel f\parallel }^{2}\le {\sum }_{i\in I}|〈f,{f}_{i}〉{|}^{2}\le B{\parallel f\parallel }^{2}$ for all $f\in ℋ$. A system $𝒱={\left(\left({V}_{i},{V}_{i}\right)\right)}_{i\in I}$ is a fusion frame or a frame of subspaces if

${A\parallel f\parallel }^{2}\le \sum _{i\in I}{v}_{i}^{2}\parallel {\pi }_{{V}_{i}}{\left(f\right)\parallel }^{2}\le B{\parallel f\parallel }^{2}$

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 ${\left(\left({S}_{𝒱}^{-1}{V}_{i},{v}_{i}\right)\right)}_{i\in I}$ (with ${S}_{𝒱}$ the frame operator given by ${\sum }_{i\in I}{v}_{i}{\pi }_{{V}_{i}}\left(f\right)$) is indeed a fusion frame. Other results deal with alternate duals, i.e., systems $𝒲={\left(\left({W}_{i},{w}_{i}\right)\right)}_{i\in I}$ so that $f={\sum }_{i\in I}{v}_{i}{w}_{i}{\pi }_{{W}_{i}}{S}_{𝒱}^{-1}{\pi }_{{V}_{i}}\left(f\right)$, and frame operators for a pair of two Bessel fusion sequences (where only the upper bound above is required to hold).

##### MSC:
 46C15 Characterizations of Hilbert spaces 42C99 Non-trigonometric Fourier analysis
##### Keywords:
Hilbert space; frame; dual frame; fusion frame