Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 081, 7 pages      arXiv:1905.08655      https://doi.org/10.3842/SIGMA.2019.081

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

Received May 22, 2019, in final form October 16, 2019; Published online October 23, 2019

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 $\sum\limits_{m=0}^{\infty}a_m m^{\ell}$ < $\infty$.

Key words: positive definite; isotropic; Hilbert sphere; Schoenberg sequences.

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