A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere

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 functions on the Hilbert sphere to be in C 2 ([0, π]), it is necessary and sufficient for its ∞-Schoenberg sequence to satisfy ∞ m=0 a m m < ∞. key words: positive definite, isotropic, Hilbert sphere, Schoenberg sequences 2010 MSC: 33B10; 33C45; 42A16; 42A82; 42C10 arXiv:1905.08655v1 [math.CA]