Hai-Zhang Zhang

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

For every odd integer $d\geq3$, a continuous function $\varphi\colon[0,\infty)\to\mathbb R$ supported in $[0,\pi]$ and isotropic positive definite on $\mathbb R^d$ remains so on $\mathbb S^d$. In even dimensions, recent work shows that this transfer fails under every prescribed positive upper bound on the support. We p...