Finite extensions of Bessel sequences

TL;DR

This paper characterizes when Bessel sequences can be extended to frames with finitely many vectors and when frames can be made Parseval frames through finite-dimensional perturbations, with additional results on frame excesses.

Contribution

It provides new characterizations of finite extensions of Bessel sequences to frames and finite perturbation conditions for frames to become Parseval frames.

Findings

Bessel sequences extendable to frames with finitely many vectors

Frames transformable into Parseval frames via finite-dimensional perturbations

Results on excesses of frames and near-Riesz bases

Abstract

The paper studies finite extensions of Bessel sequences in infinite-dimensional Hilbert spaces. We provide a characterization of Bessel sequences that can be extended to frames by adding finitely many vectors. We also characterize frames that can be converted to Parseval frames by finite-dimensional perturbations. Finally, some results on excesses of frames and near-Riesz bases are derived.