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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.11187 |
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| _version_ | 1866910124640043008 |
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| author | Feng, Han Ge, Yan |
| author_facet | Feng, Han Ge, Yan |
| contents | In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from $\RR^d$ to $\sph$ and the converse. For $d=2$, it is proved that a function defined by $$f_{\t,δ}(t)=(\t-t)_+^δ, \quad δ\geq \f{d+1}2 $$ is positive definite on the unit sphere $\mathbb{S}^2$ by restricting $\t$ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11187 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isotropic Positive Definite Functions on Spheres Feng, Han Ge, Yan Classical Analysis and ODEs In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from $\RR^d$ to $\sph$ and the converse. For $d=2$, it is proved that a function defined by $$f_{\t,δ}(t)=(\t-t)_+^δ, \quad δ\geq \f{d+1}2 $$ is positive definite on the unit sphere $\mathbb{S}^2$ by restricting $\t$ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres. |
| title | Isotropic Positive Definite Functions on Spheres |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.11187 |