A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere
In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210307 |
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| Cite this: | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere / J. Jäger // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-210307 |
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Jäger, J. 2025-12-05T09:30:55Z 2019 A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere / J. Jäger // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33B10; 33C45; 42A16; 42A82; 42C10 arXiv: 1905.08655 https://nasplib.isofts.kiev.ua/handle/123456789/210307 https://doi.org/10.3842/SIGMA.2019.081 In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary and sufficient for its ∞ Schoenberg sequence to satisfy ∑ₘ₌₀ ∞ aₘmˡ < ∞. The author was a post-doctoral fellow funded by Justus Liebig University during the development of this research. I would like to express my gratitude to Professor M. Buhmann for his helpful comments on the paper. Thanks are also due to the anonymous referees for their thorough advice on how to improve this note. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| spellingShingle |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere Jäger, J. |
| title_short |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_full |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_fullStr |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_full_unstemmed |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_sort |
note on the derivatives of isotropic positive definite functions on the hilbert sphere |
| author |
Jäger, J. |
| author_facet |
Jäger, J. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary and sufficient for its ∞ Schoenberg sequence to satisfy ∑ₘ₌₀ ∞ aₘmˡ < ∞.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210307 |
| citation_txt |
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere / J. Jäger // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
| work_keys_str_mv |
AT jagerj anoteonthederivativesofisotropicpositivedefinitefunctionsonthehilbertsphere AT jagerj noteonthederivativesofisotropicpositivedefinitefunctionsonthehilbertsphere |
| first_indexed |
2025-12-07T21:25:06Z |
| last_indexed |
2025-12-07T21:25:06Z |
| _version_ |
1850886278925516800 |