Controlled scaling of Hilbert space frames for R^2

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 $R^n$.