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

## TL;DR

This paper derives a recursive formula for derivatives of isotropic positive definite functions on the Hilbert sphere and proves a conjecture linking smoothness to the decay of the Schoenberg sequence.

## Contribution

It introduces a recursive formula for derivatives and proves a conjecture relating function smoothness to Schoenberg sequence decay on the Hilbert sphere.

## Key findings

- Recursive formula for derivatives of isotropic positive definite functions
- Proof of the conjecture relating smoothness to Schoenberg sequence decay
- Characterization of $C^{2oldsymbol{ ext{ extit{ extbf{ extsc{ extcolor{white}{ extbf{.}}}}}}}(oldsymbol{ ext{ extit{ extbf{ extsc{ extcolor{white}{ extbf{.}}}}}})}$ regularity via Schoenberg sequences

## 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 Tr\"ubner 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 $\sum\limits_{m=0}^{\infty}a_m m^{\ell}<\infty$.