Some Properties of Approximately Dual Frames in Hilbert Spaces

Abstract

In the present paper, some sufficient and necessary conditions for two frames Φ = (ϕ n ) n and Ψ = (ψ n ) n under which they are approximately or generalized dual frames are determined depending on the properties of their analysis and synthesis operators. We also give a new characterization for approximately dual frames associated with a given frame and given operator by using of bounded operators. Among other things, we prove that if two frames Φ = (ϕ n ) n and Ψ = (ψ n ) n are close to each other, then we can find approximately dual frames Φ ad = (ϕ ad n ) n and Ψ ad = (ψ ad n ) n of them which are close to each other and T Φ U Φ ad = T Ψ U Ψ ad , where T Φ and T Ψ (resp. U Φ ad and U Ψ ad ) are the analysis operators (resp. synthesis operators) of the frames Φ and Ψ (resp. Φ ad and Ψ ad ), respectively. We then give some consequences on generalized dual frames. Finally, we apply these results to find some construction results for approximately dual frames for a given Gabor frame.Mathematics Subject Classification: Primary: 42C15; Secondary: 47A58.

The effect of perturbations of frames on their alternate and approximately dual frames

“…Specifically, reconstruction of the original vector from frame is typically achieved by using a so-called duality notion. A number of variations and generalizations of duality notion can be found in Li and Ogawa 2,3 in the more general context of pseudo-duality, atomic system for subspace, 4,5 approximate duality, 6,7 and generalized duality. [7][8][9] In many situations, it is important to know which properties of frames are stable if we slightly modify the elements of the systems.…”

“…Second, we show that if we do a sufficiently small perturbation of a frame in the sense of Chen et al and Christensen, 12,13 then their specific approximately dual frames are close to each other. In contrast to the work, 7 we consider the problem in a much more general setting. More precisely, we removed the imposed condition T Φ U Φ ad = T Ψ U Ψ ad , and particularly, we estimate for these cases the deviation from perfect reconstruction.…”

“…They call two Bessel sequences Φ and Φ ad approximately dual frames with approximation rate 𝜀 whenever ||Id  − T Φ U Φ ad || ≤ 𝜀 < 1. Later on, Dehghan and Hasankhani-Fard, 8 stated in Javanshiri7 as well as earlier in Holschneider,17 introduced and studied the notion of generalized duality for frames in Hilbert spaces. Recall from Javanshiri 7, Remark 2.8(i) that two frames Φ and Φ gd are generalized dual frames, if T Φ U Φ gd is just invertible.It is shown in Javanshiri7 that approximately dual frames of Φ are precisely the sequences…”

Invertibility of generalized g-frame multipliers in Hilbert spaces

“…In recent years there has been shown considerable interest by functional analysts in the study of Bessel multipliers as a generalization of the frame operators, approximately dual frames [7], generalized dual frames [7,Remark 2.8(ii)] and atomic systems for subspaces [10]. In fact, the study of this class of operators leads us to new results concerning dual frames and local atoms, two concepts at the core of frame theory.…”

Multipliers for von Neumann–Schatten Bessel sequences in separable Banach spaces

“…Following[20, Remark 2.8], we say that the HS-frame G gd = {G gd i } is a generalized dual HS-frame of G, if T G U G gd is invertible. It is noteworthy that with an argument similar to the proof of [20, Theorem 2.1] and [1, Theorem 3.1] one can show that the generalized dual HS-frames of G are precisely the sequences G gd such that…”