黎曼对称空间上核的普适性
Universality of kernels on Riemannian symmetric spaces
AI总结：
该研究刻画了黎曼对称空间上连续正定不变核的普适性，对紧与非紧对称空间分别给出等价条件，并提供实例说明普适核的构造准则。
AI中文摘要：
我们研究黎曼对称空间上连续、正定不变核的普适性性质，为紧和非紧情形提供了统一的调和分析刻画。在紧对称情形下，我们证明连续、正定不变核为C-普适当且仅当其所有球系数均严格为正，这是对经典博赫纳型结果的改进；该刻画被推广至紧齐性空间，其中普适性等价于表示论展开所得系数矩阵的严格正定性。与之相对，对于非紧对称空间，我们证明任何连续、正定且C₀可积的不变核自动为C₀-普适。我们的分析本质上依赖于利用球函数和群傅里叶变换的谱分解，以提供普适性条件的替代调和分析表述；多个例子（包括球、紧李群上的核，以及双曲空间和对称锥上的可积核）既阐释了该理论，也为构造普适核提供了实用准则。
英文摘要：
We investigate universality properties of continuous, positive-definite invariant kernels on Riemannian symmetric spaces, providing a unified harmonic-analytic characterization across compact and non-compact settings. In the compact symmetric case, we prove that a continuous, positive-definite invariant kernel is $C$-universal if and only if all of its spherical coefficients are strictly positive, a sharpening of classical Bochner-type results. This characterization is extended to compact homogeneous spaces, where universality is shown to be equivalent to the strict positive-definiteness of coefficient matrices arising from a representation-theoretic expansion. In contrast, for non-compact symmetric spaces, we establish that any continuous, positive-definite invariant kernel that is $C_0$ and integrable is automatically $C_0$-universal. Our analysis essentially relies on spectral decompositions using spherical functions and group Fourier transforms, in order to provide alternative, harmonic-analytic formulations of universality conditions. Several examples (including kernels on spheres and compact Lie groups, as well as integrable kernels on hyperbolic spaces and symmetric cones) both illustrate the theory and demonstrate practical criteria for constructing universal kernels.
