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A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere
Janin Jäger Lehrstuhl Numerische Mathematik, Justus-Liebig University, Heinrich-Buff Ring 44, 35392 Giessen, Germany
Abstract:
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 $C^{2\ell}([0,\pi])$, it is necessary and sufficient for its $\infty$-Schoenberg sequence to satisfy $\sum\limits_{m=0}^{\infty}a_m m^{\ell}<\infty$.
Keywords:
positive definite, isotropic, Hilbert sphere, Schoenberg sequences.
Received: May 22, 2019; in final form October 16, 2019; Published online October 23, 2019
Citation:
Janin Jäger, “A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere”, SIGMA, 15 (2019), 081, 7 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1517 https://www.mathnet.ru/eng/sigma/v15/p81
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Abstract page: | 99 | Full-text PDF : | 18 | References: | 10 |
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