Dynamical sampling and frame representations with bounded operators

Abstract

The purpose of this paper is to study frames for a Hilbert space $\mathcal{H},$ having the form $\{T^n\phi {\}}_{n=0}^{\infty }$ for some $\phi \in \mathcal{H}$ and an operator $T:\mathcal{H}\to \mathcal{H}.$ We characterize the frames that have such a representation for a bounded operator $T,$ and discuss the properties of this operator. In particular, we prove that the image chain of $T$ has finite length $N$ in the overcomplete case; furthermore $\{T^n\phi {\}}_{n=0}^{\infty }$ has the very particular property that $\{T^n\phi {\}}_{n=0}^{N-1}\cup \{T^n\phi {\}}_{n=N+\ell }^{\infty }$ is a frame for $\mathcal{H}$ for all $\ell \in {\mathbf{N}}_0$. We also prove that frames of the form $\{T^n\phi {\}}_{n=0}^{\infty }$ are sensitive to the ordering of the elements and to norm-perturbations of the generator $\phi$ and the operator $T.$ On the other hand positive stability results are obtained by considering perturbations of the generator $\phi$ belonging to an invariant subspace on which $T$ is a contraction.

Accepted for publication in J. Math. Anal. Appl