Related papers: Dynamical sampling: mixed frame operators, represe…
Dynamical sampling deals with representations of a frame $\{ f_k \}_{k=1}^\infty$ as an orbit $\{ T^n \varphi \}_{n=0}^\infty$ of a linear and possibly bounded operator $T$ acting on the underlying Hilbert space. It is known that the desire…
The purpose of this paper is to study frames for a Hilbert space ${\cal H},$ having the form $\{T^n \varphi\}_{n=0}^\infty$ for some $\varphi \in {\cal H}$ and an operator $T: {\cal H} \to {\cal H}.$ We characterize the frames that have…
Motivated by recent work in Dynamical Sampling, we prove a necessary and sufficient condition for a frame in a separable and infinite-dimensional Hilbert space to admit the form $\{T^{n} \varphi \}_{n \geq 0}$ with $T \in B(H)$. Also, a…
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…
This paper is a contribution to the theory of dynamical sampling. Our purpose is twofold. We first consider representations of sequences in a Hilbert space in terms of iterated actions of a bounded linear operator. This generalizes recent…
Recent research has shown that the properties of overcomplete Gabor frames and frames arising from shift-invariant systems form a precise match with certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a…
Dynamical sampling deals with frames of the form $\{T^n\varphi\}_{n=0}^\infty$, where $T \in B(\mathcal{H})$ belongs to certain classes of linear operators and $\varphi\in\mathcal{H}$. The purpose of this paper is to investigate a new…
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…
The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbf Z}$ having the form $\{T^kf_0\}_{k\in\mathbf Z}$ for some linear operator $T: \mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}$. A key result…
It is known that it is a very restrictive condition for a frame $\{f_k\}_{k=1}^\infty$ to have a representation $ \{T^n \varphi\}_{n=0}^\infty$ as the orbit of a bounded operator $T$ under a single generator $\varphi\in\mathcal{H}.$ In this…
In this paper, we give a multiplication operator representation of bounded self-adjoint operators T on a Hilbert space H such that -- is a frame for H, for some -- . We state a necessary condition in order for a frame -- to have a…
We review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form $\{A^ng: g\in G,\, n=0,1,2,\dots \}$, where $A$ is a bounded linear operators on a separable complex…
Motivated by the dynamical sampling problem, we study frames in an infinite dimensional Hilbert space generated by the iterates of a bounded operator T, also known as dynamical frames. We first characterize the operators that generate…
To be able to solve operator equations numerically a discretization of those operators is necessary. In the Galerkin approach bases are used to achieve discretized versions of operators. In a more general set-up, frames can be used to…
We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two…
One of the most important problems in the studying of frames and its extensions is the invariance of these systems under perturbation. The current paper is concerned with the invariance of Modular biframes for operators under some class of…
We consider bounded operators $A$ acting iteratively on a finite set of vectors $\{f_i : i\in I\}$ in a Hilbert space $\mathcal H$ and address the problem of providing necessary and sufficient conditions for the collection of iterates…
Physical motions are inherently continuous, and higher camera frame rates typically contribute to improved smoothness and temporal coherence. For the first time, we explore continuous representations of human motion sequences, featuring the…
In this note, we solve the dynamical sampling problem for a class of shift-preserving operators $L:V\to V$ acting on a finitely generated shift-invariant space $V$. We find conditions on $L$ and a finite set of functions of $V$ so that the…
Frames formed by orbits of vectors through the iteration of a bounded operator have recently attracted considerable attention, in particular due to its applications to dynamical sampling. In this article, we consider two commuting bounded…