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

A system (f i ) iI is a frame in a Hilbert space if there are positive constants A and B such that Af 2 iI |f,f i | 2 Bf 2 for all f. A system 𝒱=((V i ,V i )) iI is a fusion frame or a frame of subspaces if

Af 2 iI v i 2 π V i (f) 2 Bf 2

where π 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 𝒱 -1 V i ,v i )) iI (with S 𝒱 the frame operator given by iI v i π V i (f)) is indeed a fusion frame. Other results deal with alternate duals, i.e., systems 𝒲=((W i ,w i )) iI so that f= iI v i w i π W i S 𝒱 -1 π 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