Controlled scaling of Hilbert space frames for R^2

TL;DR

This paper investigates how to optimally scale vectors in Hilbert space frames for R^2 to minimize their condition number, aiming to make frames as tight as possible within specified scaling constraints.

Contribution

It provides the first analysis of optimal scaling strategies in R^2 for improving frame tightness and condition number minimization under bounded scaling factors.

Findings

Derived methods for scaling frames to minimize condition number in R^2

Established bounds for scaling factors within [1-ε, 1+ε]

Paved the way for extending results to higher dimensions

Abstract

A Hilbert space frame on $R^n$ is {\it scalable} if we can scale the vectors to make them a tight frame. There are known classifications of scalable frames. There are two basic questions here which have never been answered in any $R^n$: Given a frame in $R^n$, how do we scale the vectors to minimize the condition number of the frame? I.e. How do we scale the frame to make it as tight as possible? If we are only allowed to use scaling numbers from the interval $[1-\epsilon,1+\epsilon]$, how do we scale the frame to minimize the condition number? We will answer these two questions in $R^2$ to begin the process towards a solution in $R^n$.