论文标题
R^2的希尔伯特空间框架的受控缩放
Controlled scaling of Hilbert space frames for R^2
论文作者
论文摘要
如果我们可以扩展向量以使其成为紧密的框架，则$ r^n $上的Hilbert空间框架是{\ it可伸缩}。有可扩展帧的已知分类。这里有两个基本问题，这些问题从未在任何$ r^n $中得到回答： 给定$ r^n $的框架，我们如何扩展向量以最大程度地减少框架的状况数？ IE。我们如何缩放框架以使其尽可能紧密？ 如果我们仅允许我们从间隔$ [1-ε，1+ε] $中使用缩放数字，我们如何缩放帧以最小化条件号？ 我们将在$ r^2 $中回答这两个问题，以开始在$ r^n $中的解决方案。
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 $[1-ε,1+ε]$, 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 $R^n$.
