Completion versus removal of redundancy by perturbation

TL;DR

This paper investigates how small perturbations can either enable complete representation of a Hilbert space or remove redundancy in sequences with expansion properties, with implications for dynamical sampling.

Contribution

It provides conditions under which small perturbations can complete or remove redundancy in sequences with expansion properties in Hilbert spaces.

Findings

Small norm-perturbations enable representation for frame and Riesz sequences.

Redundancy can be removed via perturbations for certain frames.

Results are motivated by recent advances in dynamical sampling.

Abstract

A sequence $\{g_k\}_{k=1}^\infty$ in a Hilbert space $\cal H$ has the expansion property if each $f\in \overline{\text{span}} \{g_k\}_{k=1}^\infty$ has a representation $f= \sum_{k=1}^\infty c_k g_k$ for some scalar coefficients $c_k.$ In this paper we analyze the question whether there exist small norm-perturbations of $\{g_k\}_{k=1}^\infty$ which allow to represent all $f\in \cal H;$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\{g_k\}_{k=1}^\infty$ such that $g_k\to 0$ as $k\to \infty,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.