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

Abstract

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 prove an exact-support refinement with a construction uniform in the prescribed radius. More precisely, for each $d=2m\geq2$ and $R\in(0,\pi]$, we construct a function $\varphi$ whose radial extension belongs to $C_c^\infty(\mathbb R^d)$ and has support radius exactly $R$, such that $\varphi(\|\mathbf{x}-\mathbf{y}\|_2)$ is strictly positive definite on $\mathbb R^d$, whereas $\varphi(\rho(\mathbf{x},\mathbf{y}))$ is not positive definite on $\mathbb S^d$. Thus every admissible support radius is attained by a smooth, strictly Euclidean positive-definite counterexample.