Finite frames are sequences of vectors in finite dimensional Hilbert spaces that play a key role in signal processing and coding theory. In this paper, we study the class of tight unit-norm frames for \(\mathbb {C}^{d}\) that also form regular schemes, which we call tight regular schemes (TRS). Many common frames that arise in applications such as equiangular tight frames and mutually unbiased bases fall in this class. We investigate characteristic properties of TRSs and prove that for many constructions, they are intimately connected to weighted 1-designs—arising from cubature rules for integrals over spheres in \(\mathbb {C}^{d}\)—with weights dependent on the Voronoi regions of each frame element. Aided by additional numerical evidence, we conjecture that all TRSs in fact satisfy this property.

中文翻译：

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作为常规方案的紧框架的属性

有限帧是有限维希尔伯特空间中向量的序列，在信号处理和编码理论中起着关键作用。在本文中，我们研究\（\ mathbb {C} ^ {d} \）的紧单位范数框架的类别，这些框架也形成了常规方案，我们称其为严格常规方案（TRS）。在应用中出现的许多常见框架，例如等角紧框架和相互无偏的基体，都属于此类。我们研究了TRS的特征，并证明了对于许多结构，它们与加权1-设计紧密相关-从\（\ mathbb {C} ^ {d} \）中球体积分的容积规则中得出-权重取决于每个框架元素的Voronoi区域。借助其他数字证据，我们推测所有的TRS实际上都满足此特性。