Fatemeh Ghobadzadeh | Searcharxiv

arXiv2020

Properties of bounded representations for $G$-frames

Due to the importance of frame representation by a bounded operator in dynamical sampling, researchers studied the frames of the form $\{T^{i-1} f\}_{i\in \mathbb{N}}$, which $f$ belongs to separable Hilbert space $\mathcal{H}$ and $T\in B(\mathcal{H})$, and investigated the properties of $T$. Given that $g$-frames include the wide range of frames such as fusion frames, the main purpose of this paper is to study the characteristics of the operator $T$ for $g$-frames of the form $\{ΛT^{i-1} \in B(\mathcal{H},\mathcal{K}):i\in \mathbb{N}\}$.

math.FA↗

arXiv2018

g-frame representations with bounded operators

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form $\{T^i f_1\}_{i\in \mathbb{N}}$, where $f_1$ belongs to Hilbert space $\h$ and $T:\h\rightarrow\h$ belongs to certain classes of the bounded operators. Christensen et al., study frames for $\h$ with index set $\mathbb{N}$ (or $\mathbb{Z}$), that have representations in the form $\{T^{i-1}f_1\}_{i\in \mathbb{N}}$ (or $\{T^if_0\}_{i\in \mathbb{Z}}$). As frames of subspaces, fusion frames and generalized translation invariant systems are the spacial cases of $g$-frames, the purpose of this paper is to study $g$-frames $Λ=\{Λ_i\in B(\h,\K): i\in I\}$ $(I=\mathbb{N}$ or $\mathbb{Z}$) having the form $Λ_{i+1}=Λ_1 T^{i},$ for $T\in B(\h).$

math.FA↗