Positive definite singular kernels on two-point homogeneous spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916459557421056 |
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| author | Bilyk, Dmitriy Grabner, Peter |
| author_facet | Bilyk, Dmitriy Grabner, Peter |
| contents | We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous
spaces. As opposed to the classical case, which has been developed and studied
in the existing literature, we allow the kernel to have an (integrable)
singularity for $x=y$. Specifically, the Riesz kernel $d(x,y)^{-s}$ (where
$d$ denotes some distance on the space) is a prominent example. We derive
results analogous to Schoenberg's characterization of positive definite
functions on the sphere, Schur's lemma on the positive definiteness of the
product of positive definite functions, and Schoenberg's characterization of
functions positive definite on all spheres. We use these results to better
understand the behavior of the Riesz kernels for the geodesic and chordal
distances on projective spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22104 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive definite singular kernels on two-point homogeneous spaces Bilyk, Dmitriy Grabner, Peter Classical Analysis and ODEs We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity for $x=y$. Specifically, the Riesz kernel $d(x,y)^{-s}$ (where $d$ denotes some distance on the space) is a prominent example. We derive results analogous to Schoenberg's characterization of positive definite functions on the sphere, Schur's lemma on the positive definiteness of the product of positive definite functions, and Schoenberg's characterization of functions positive definite on all spheres. We use these results to better understand the behavior of the Riesz kernels for the geodesic and chordal distances on projective spaces. |
| title | Positive definite singular kernels on two-point homogeneous spaces |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.22104 |