Dynamical sampling and frame representations with bounded operators

The purpose of this paper is to study frames for a Hilbert space H, having the form {T-n phi},:(infinity)(n=0) for some phi is an element of H and an operator T : H -> H We characterise 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)(infinity) has the very particular property that {T-n phi}(n=0)(N-1) U {Tn phi}(n=N+l)(infinity) is a frame for for all l is an element of N-0. We also prove that frames of the form {T-n phi}(n=0)(infinity) 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. (C) 2018 Elsevier Inc. All rights reserved.