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 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 < ∞.