距离几何中的正定函数
Positive definite functions in distance geometry
摘要：
I.勋伯格证明了一个函数在单位球面上是正定的当且仅当这个函数是盖根堡多项式的非负线性组合。这一事实发挥了至关重要的作用，德尔萨特的方法寻找边界的密度球包装领域和欧几里得空间。德尔萨特方法最令人兴奋的应用之一是解决8维和24维的接吻数问题。然而，8和24是该方法给出精确结果的唯一维度。对于其他维度（例如，三维和四维），上界超过下限。我们已经找到了Delsarte方法的一个扩展，它可以解决三维和四维的接吻数问题（以及单侧接吻数问题）。本文还讨论了球面二距离集的极大基数。使用所谓的多项式方法和Delsarte方法，可以确定所有维度$n<40$的这些基数。最近，Schoenberg定理被推广到多元正定函数。利用这些推广和半定规划可以改进球码的一些上界。
英文摘要：
I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsarte's method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of the Delsarte method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions three and four. In this paper we also will discuss the maximal cardinalities of spherical two-distance sets. Using the so-called polynomial method and Delsarte's method these cardinalities can be determined for all dimensions $n<40$. Recently, were found extensions of Schoenberg's theorem for multivariate positive-definite functions. Using these extensions and semidefinite programming can be improved some upper bounds for spherical codes.
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