Welch边界的几何
Geometry of the Welch Bounds
摘要：
一个几何的角度，涉及Grammian和框架运营商是用来获得整个家庭的Welch界限。这个观点统一了一些观察，已经取得了关于紧的边界和它们的连接对称k-张量，紧框架，齐次多项式，和t-设计。尤其是。在齐次多项式的采样和对称k张量的框架之间已经建立了联系。它还表明，紧的界限需要紧的框架。然而，在许多情况下，对称k张量缺乏紧框架，导致考虑尽可能接近达到边界的集合。几何推导，然后扩展到广义或连续框架的设置。有限集和可数无限集的韦尔奇界成为这个一般设置的特殊情况。
英文摘要：
A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular. a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames in symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
