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 $C^{2\ell }([0,\pi ])$, it is necessary and sufficient for its $\infty$-Schoenberg sequence to satisfy $\sum _{m=0}^{\infty }{a}_m{m}^{\ell }<\infty$.

References in corpus (3)

• Schoenberg's theorem for real and complex Hilbert spheres revisited
• Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions
• Thinplate Splines on the Sphere