Research PaperNo Access

$G$ -frame representations with bounded operators

Yavar Khedmati

Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran

E-mail Address: khedmati.y@uma.ac.ir

E-mail Address: khedmatiy.y@gmail.com

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and

Fatemeh Ghobadzadeh

Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran

E-mail Address: gobadzadehf@yahoo.com

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https://doi.org/10.1142/S0219691320500782Cited by:2 (Source: Crossref)

Abstract

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form ${T^{i - 1} f_{1}}_{i \in ℕ}$ , where $f_{1}$ belongs to Hilbert space $ℋ$ and $T : ℋ \to ℋ$ belongs to certain classes of bounded operators. Christensen et al. studied frames for $ℋ$ with index set $ℕ$ (or $ℤ$ ), that has representations in the form ${T^{i - 1} f_{1}}_{i \in ℕ}$ (or ${T^{i} f_{0}}_{i \in ℤ}$ ). As frames of subspaces, fusion frames and generalized translation invariant systems are the special cases of $g$ -frames, the purpose of this paper is to study and get sufficient conditions for $g$ -frames $Λ = {Λ_{i} \in B (ℋ, 𝒦) : i \in ℕ$ (or $ℤ)}$ having the form $Λ_{i + 1} = Λ_{1} T^{i},$ $T \in B (ℋ)$ (or $Λ_{i + 1} = Λ_{0} T^{i},$ $T \in G L (ℋ)$ ). Also, we get the relation between representations of dual $g$ -frames with index set $ℤ$ . Finally, we study stability of $g$ -frame representations under some perturbations.

Keywords:

AMSC: 41A58, 42C15, 47A05

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