Dynamical sampling: mixed frame operators, representations and perturbations

TL;DR

This paper explores the structure, representation, and perturbation of frames generated by powers of bounded linear operators, advancing the understanding of dynamical sampling and mixed frame operators.

Contribution

It introduces new results on representing frames via operator iterations, analyzes perturbations, and extends dynamical sampling theory with scalar-weighted frame sequences.

Findings

Frames of the form {T^n φ} can be characterized and represented using operator iterations.

Perturbation results provide stability conditions for sequences {T^n φ}.

Scalar-weighted frames {a_n T^n φ} are effectively analyzed within the dynamical sampling framework.

Abstract

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}$.