Subfusion Frames

Abstract

Fusion frames are generalizations of frames in Hilbert spaces which were introduced by Casazza et al. (2008). In the present paper, we study the relations between fusion frames and subfusion frame operators. Specially, we introduce new construction of subfusion frames and derive new results.

Some properties of alternate duals and approximate alternate duals of fusion frames

“…Indeed, take W 1 = span{(1, 0, 0)}, W 2 = span{(1, 1, 0)}, W 3 = span{(0, 1, 0)}, W 4 = span{(0, 0, 1)}, and ω 1 = ω 3 = ω 4 = 1, ω 2 = √ 2. Then W = {(W i , ω i )} i∈I is a fusion frame for R 3 with an alternate dual as V = {(V i , υ i )} i∈I where V 1 = span{(0, 1, 0)}, V 2 = R 3 , V 3 = span{(1, 0, 0)}, V 4 = span{(0, 0, 1)}, and υ 1 = υ 3 = 3 , υ 2 = 3 √ 2, υ 4 = 1; see Example 3.1 of [1]. A straightforward calculation shows that W is not an alternate dual fusion frame of V .…”

Some properties of dual and approximate dual of fusion frames

“…where 1 2 ≤ α < 1 and 0 ≤ β ≤ 1 100 , then U = {U i } i∈I is a ǫ-perturbation of V with ǫ < 1 8 . Hence, by Theorem 3.2, U is also an approximate alternate dual fusion frame of W .…”

“…Indeed, take W 1 = span{(1, 0, 0)}, W 2 = span{(1, 0)}, W 3 = span{(0, 1, 0)}, W 4 = span{(0, 0, 1)}, and ω 1 = ω 3 = ω 4 = 1, ω 2 = √ 2. Then W = {(W i , ω i )} i∈I is a fusion frame for R 3 with an alternate dual as V = {(V i , υ i )} i∈I where V 1 = span{(0, 1, 0)}, V 2 = R 3 , V 3 = span{(1, 0, 0)}, V 4 = span{(0, 0, 1)}, and υ 1 = υ 3 = 3 , υ 2 = 3 √ 2, υ 4 = 1, see Example 3.1 of [1]. A straightforward calculation shows that W is not an alternate dual fusion frame of V .…”