Dynamical sampling: mixed frame operators, representations and perturbations

Abstract

Motivated by recent progress in operator representation of frames, we investigate the frames of the form {T n ϕ} n∈I for I = N, Z, and answer questions about representations, perturbations and frames induced by the action of powers of bounded linear operators. As a particular case, we discuss problems concerning representation of frames in terms of iterations of the mixed frame operators. As our another contribution, we consider frames of the form {a n T n ϕ} ∞ n=0 for some non-zero scalars {a n } ∞ n=0 , and we obtain some new results in dynamical sampling. Finally, we will present some auxiliary results related to the perturbation of sequences of the form {T n ϕ} ∞ n=0 .2010 Mathematics Subject Classification. 42C15, 47B40.

Fibonacci representations of sequences in Hilbert spaces

“…In [2], Aldroubi et al introduced dynamic sampling which it deals with frame properties of sequences of the form {T n ϕ} ∞ n=0 , where T ∈ B(H) belongs to certain classes of linear operators (such as diagonalizable normal operators) and ϕ ∈ H. Various characterizations of frames having the form {f k } k∈I = {T k ϕ} k∈I , where T is a linear (not necessarily bounded) operator can be found in [5,6,7,8].…”