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

Abstract

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 ${\mathbb{S}}^d$. In even dimensions, recent work shows that this transfer fails under every prescribed positive upper bound on the support. We prove an exact-support refinement with a construction uniform in the prescribed radius. More precisely, for each $d=2m\ge 2$ and $R\in (0,\pi ]$, we construct a function $\phi$ whose radial extension belongs to $C_c^{\infty }({\mathbb{R}}^d)$ and has support radius exactly $R$, such that $\phi (\Vert \mathbf{x}-\mathbf{y}{\Vert }_2)$ is strictly positive definite on ${\mathbb{R}}^d$, whereas $\phi (\rho (\mathbf{x},\mathbf{y}))$ is not positive definite on ${\mathbb{S}}^d$. Thus every admissible support radius is attained by a smooth, strictly Euclidean positive-definite counterexample.

arXiv comment14 pages; minor expository revisions and clarifications, updated MSC classification, and added an acknowledgements section; results unchanged