论文标题
四季度投影球形设计的差异表征
A variational characterisation of projective spherical designs over the quaternions
论文作者
论文摘要
我们对Quaternionic Hilbert Space $ \ hd $的矢量/线的包装不平等，该$ \ hd $概括了Sidelnikov和Welch的单位矢量，以$ \ rd $和$ \ cd $。这具有一个参数$ t $，并且仅取决于向量，直到投影统一的等效性。 $ {\ Mathbb {f}}^d = {\ Mathbb {r}}^d，{\ Mathbb {c}}^d，{\ Mathbb {\ Mathbb {h}^d $ cubs cub（cub）的序列，在$ {\ mathbb {f}}^d = = {\ Mathbb {r}}^d，{ $ {\ mathbb {f}}^d $用于合适的多项式空间$ \ hom _ {\ fd}（t，t，t）$。使用此情况，我们表明Delsarte空间上的投射球形$ t $ -Designs $ \ ff p^{d-1} $与球形$（t，t）$ - 单位矢量设计的$ {\ Mathbb {f}}}^d $的单位矢量设计。然后，我们探讨了Quaternionic空间中的许多示例。单位不变的多项式空间$ {\ mathop {\ rm hom} \ nolimits} _ {\ mathbb {h}^d}（t，t，t）$和我们在其上定义的内部产品，因此复制kernel具有简单的形式。
We give an inequality on the packing of vectors/lines in quaternionic Hilbert space $\Hd$, which generalises those of Sidelnikov and Welch for unit vectors in $\Rd$ and $\Cd$. This has a parameter $t$, and depends only on the vectors up to projective unitary equivalence. The sequences of vectors in ${\mathbb{F}}^d={\mathbb{R}}^d,{\mathbb{C}}^d,{\mathbb{H}}^d$ that give equality, which we call spherical $(t,t)$-designs, are seen to satisfy a cubature rule on the unit sphere in ${\mathbb{F}}^d$ for a suitable polynomial space $\Hom_{\Fd}(t,t)$. Using this, we show that the projective spherical $t$-designs on the Delsarte spaces $\FF P^{d-1}$ coincide with the spherical $(t,t)$-designs of unit vectors in ${\mathbb{F}}^d$. We then explore a number of examples in quaternionic space. The unitarily invariant polynomial space ${\mathop{\rm Hom}\nolimits}_{\mathbb{H}^d}(t,t)$ and the inner product that we define on it so the reproducing kernel has a simple form are of independent interest.
