A Sharpening of the Welch Bounds and the Existence of Real and Complex Spherical $t$ –Designs
A Sharpening of the Welch Bounds and the Existence of Real and Complex Spherical $t$ –Designs
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韦尔奇界限的锐化以及真实和复杂球形 $t$ 的存在 – 设计
DOI：
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发表时间：
2017
影响因子：
2.5
通讯作者：
S. Waldron
中科院分区：
文献类型：
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作者：
S. Waldron
单位向量的有限集的Welch界是一个由<inline-formula><tex-math notation="LaTeX">$t=1，2，ldots $</tex-math></inline-formula>索引的不等式族，它描述了向量如何“均匀分布”。它们在信号分析中有重要的应用，其中在第一Welch界中给出相等的序列被称为Welch界相等序列或单位范数紧框架。在这里，我们考虑序列的向量给予平等的高阶韦尔奇界限。这些被认为是对应于紧框架的复杂的对称<inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula>-张量（我们证明总是存在的）。我们表明，对于<inline-formula><tex-math notation="LaTeX">$t>1$</tex-math></inline-formula>，韦尔奇界可以锐化为真实的向量，并再次，向量给予平等总是存在的。我们给出了一个统一的处理各种条件的平等在真实的和复杂的情况。特别是，我们给出了一个明确的描述相应的容积规则（<inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula>-设计）。我们的研究结果建立了一个框架的建设和分类最近感兴趣的几种配置的向量。这些包括相互无偏基地，复杂的等角线，球面半设计，投影<inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula>-设计，和最小的高阶框架潜力。一个有趣的结果是一组复杂的等角线，这是以前未知的建设。
The Welch bounds for a finite set of unit vectors are a family of inequalities indexed by <inline-formula> <tex-math notation="LaTeX">$t=1,2,ldots $ </tex-math></inline-formula>, which describe how “evenly spread” the vectors are. They have important applications in signal analysis, where sequences giving equality in the first Welch bound are known as Welch bound equality sequences or as unit norm tight frames. Here, we consider sequences of vectors giving equality in the higher order Welch bounds. These are seen to correspond to tight frames for the complex symmetric <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>–tensors (which we prove always exist). We show that for <inline-formula> <tex-math notation="LaTeX">$t>1$ </tex-math></inline-formula>, the Welch bounds can be sharpened for real vectors, and again, vectors giving equality always exist. We give a unified treatment of various conditions for equality in both the real and complex cases. In particular, we give an explicit description of the corresponding cubature rules (<inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>–designs). Our results set up a framework for the construction and classification several configurations of vectors of recent interest. These include mutually unbiased bases, complex equiangular lines, spherical half–designs, projective <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>–designs, and minimisers of the higher order frame potential. One interesting consequence is a construction of sets of complex equiangular lines, which were previously unknown.