TAILIEUCHUNG - Some properties of alternate duals and approximate alternate duals of fusion frames
In this paper we extend the notion of approximate dual to fusion frames and present some approaches to obtain alternate dual and approximate alternate dual fusion frames. We also study the stability of alternate dual and approximate alternate dual fusion frames. | Turk J Math (2017) 41: 1191 – 1203 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Some properties of alternate duals and approximate alternate duals of fusion frames Ali Akbar AREFIJAMAAL, Fahimeh ARABYANI NEYSHABURI∗ Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we extend the notion of approximate dual to fusion frames and present some approaches to obtain alternate dual and approximate alternate dual fusion frames. We also study the stability of alternate dual and approximate alternate dual fusion frames. Key words: Fusion frames, alternate dual fusion frames, approximate alternate duals, Riesz fusion bases 1. Introduction and preliminaries Fusion frame theory is a natural generalization of frame theory in separable Hilbert spaces, introduced by Casazza and Kutyniok in [4]. Fusion frames are applied to signal processing, image processing, sampling theory, filter banks, and a variety of applications that cannot be modeled by discrete frames [11, 14]. Let I be a countable index set and recall that a sequence {fi }i∈I is a frame in a separable Hilbert space H if there exist constants 0 0 , i ∈ I . Then {(Wi , ωi )}i∈I is called a fusion frame for H if there exist the constants 0 0. Also let {fi,j }j∈Ji be a frame for Wi with frame bounds αi and βi such that 0 < α = infi∈I αi ≤ β = supi∈I βi < ∞. Then the following conditions are equivalent: (i) {(Wi , ωi )}i∈I is a fusion frame of H with bounds C and D . (ii) {ωi fi,j }i∈I,j∈Ji is a frame of H with bounds αC and βD . 1192 () AREFIJAMAAL and ARABYANI NEYSHABURI/Turk J Math Recall that for each sequence {Wi }i∈I of closed subspaces in H , the space ∑ ⊕Wi = {{fi }i∈I : fi ∈ Wi , i∈I ∑ ∥fi ∥2 < ∞}, i∈I with the inner product ⟨{fi }i∈I , {gi }i∈I ⟩ = ∑ ⟨fi , gi .
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