SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 17-57, January 2021.
Kernel matrices are of central importance to many applied fields. In this manuscript, we focus on spectral properties of kernel matrices in the so-called “flat limit,” which occurs when points are close together relative to the scale of the kernel. We establish asymptotic expressions for the determinants of the kernel matrices, which we then leverage to obtain asymptotic expressions for the main terms of the eigenvalues. Analyticity of the eigenprojectors yields expressions for limiting eigenvectors, which are strongly tied to discrete orthogonal polynomials. Both smooth and finitely smooth kernels are covered, with stronger results available in the finite smoothness case.

中文翻译：

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平坦极限中的核矩阵的光谱特性

SIAM矩阵分析和应用杂志，第42卷，第1期，第17-57页，2021年1月。
内核矩阵对于许多应用领域都至关重要。在本手稿中，我们将重点放在内核矩阵的光谱特性上，即所谓的“平坦极限”，即当点相对于内核的尺度彼此靠近时发生。我们为核矩阵的行列式建立渐近表达式，然后利用它们来获取特征值主要项的渐近表达式。本征投影仪的解析性产生了用于限制本征向量的表达式，这些表达式与离散的正交多项式紧密相关。涵盖了平滑和有限平滑的内核，在有限平滑情况下可获得更强的结果。