Abstract

Schoenberg’s classical 1938 theorem asserts that, given a function \(\rho\colon G\times G\to\mathbb{C}\), the function \(\exp(-t\rho)\) is a positive definite kernel on \(G\times G\) for any \(t>0\) if and only if the kernel \(\rho\) is Hermitian and negative definite on \(G\times G\). An analog of this theorem for matrices was essentially proved by C. Löwner in 1966. Recently (in 2021), C. Dörr and M. Schlather obtained an analog of Schoenberg’s theorem for real matrix-valued functions \(\rho(x)\), \(x\in \mathbb{R}^d\). This analog refers to conditionally negative definite matrix-valued functions. In the present paper, \(a\)-conditionally negative definite matrix-valued kernels \(\rho\) on \(G\times G\) for which an analog of Schoenberg’s theorem holds are introduced and studied. The following more general problem is also considered: for what functions \(f\) and \(g\) and matrix-valued kernels \(\rho\) on \(G\times G\) is the function \(f(tg(\rho))\) a positive definite matrix-valued kernel on \(G\times G\) for any \(t>0\)? Necessary conditions, sufficient conditions, and examples of such functions are given.

中文翻译：

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$$a$$ 的勋伯格定理的模拟 - 条件负定矩阵值核

摘要

勋伯格经典 1938 定理断言，给定函数\(\rho\colon G\times G\to\mathbb{C}\)，函数\(\exp(-t\rho)\)是一个正定核对于任何\(t>0\)，\(G\times G\)当且仅当核\(\rho\)是 Hermitian 且在\(G\times G\)上负定。C. Löwner 在 1966 年基本上证明了该矩阵定理的类比。最近（2021 年），C. Dörr 和 M. Schlather 获得了实矩阵值函数 Schoenberg 定理的类比 \(\rho(x) \ ) , \(x\in \mathbb{R}^d\)。该类比指的是条件负定矩阵值函数。在本论文中，\(a\) -在\(G\times G\)上的条件负定矩阵值核\(\rho\)，其勋伯格定理的模拟成立，并进行了介绍和研究。还考虑以下更普遍的问题：对于什么函数\(f\)和\(g\)以及\(G\times G\)上的矩阵值核\(\rho\)来说，函数\(f( tg(\rho))\)对于任何\(t>0\)的\(G\times G\)上的正定矩阵值核？给出了此类函数的必要条件、充分条件和示例。