Frames containing a Riesz basis and preservation of this property under perturbation

TL;DR

This paper investigates the structure and stability of frames containing a Riesz basis, examining how perturbations affect their overcompleteness and providing a construction of a frame without a Riesz basis.

Contribution

It characterizes overcompleteness of frames via properties of bounded operators and analyzes the impact of different perturbation types on frame properties.

Findings

Overcompleteness is preserved under certain compact operator perturbations.

Paley-Wiener perturbations do not preserve overcompleteness.

A frame without a Riesz basis is constructed based on unconditional convergence issues.

Abstract

Aldroubi has shown how one can construct any frame $\gtu$ starting with one frame $\ftu $,using a bounded operator $U$ on $l^2(N)$. We study the overcompleteness of the frames in terms of properties of $U$. We also discuss perturbation of frames in the sense that two frames are ``close'' if a certain operator is compact. In this way we obtain an equivalence relation with the property that members of the same equivalence class have the same overcompleteness. On the other hand we show that perturbation in the Paley-Wiener sense does not have this property. \\ Finally we construct a frame which is norm-bounded below, but which does not contain a Riesz basis.The construction is based on the delicate difference between the unconditional convergence of the frame representation, and the fact that a convergent series in the frame elements need not converge unconditionally.