Exact-Support Counterexamples to Euclidean-to-Spherical Transfer of Positive Definiteness in Even Dimensions

Wentao Huang, Haizhang Zhang

For every odd integer $d\ge 3$, a continuous function $\phi :[0,\infty )\to \mathbb{R}$ supported in $[0,\pi ]$ and isotropic positive definite on ${\mathbb{R}}^d$ remains so on $\mathbb S…