Dynamical sampling: mixed frame operators, representations and perturbations

Motivated by recent progress in operator representation of frames, we investigate the frames of the form $\{T^n \varphi\}_{n\in I}$ for $I=\mathbb{N}, \mathbb{Z}$, 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 $\{a_n T^n \varphi\}_{n=0}^{\infty}$ for some non-zero scalars $\{a_n\}_{n=0}^{\infty}$, 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 $\{T^n \varphi\}_{n=0}^{\infty}$.