AI 中文总结
该研究针对偶数维，构造了支撑半径精确为给定值的光滑函数，证明其在欧氏空间严格正定但在球面不正定，给出了正定性传递失败的精确支撑反例。
AI 中文摘要
对于每个奇数整数d≥3，支撑在[0,π]上且在ℝ^d上各向同性正定的连续函数φ:[0,∞)→ℝ，在球面𝕊^d上仍保持各向同性正定。对于偶数维，近期研究表明，在对支撑集的任意给定正上界下，该传递均不成立。我们证明了一个精确支撑的改进结果，其构造对给定半径具有一致性。更确切地说，对于每个d=2m≥2及R∈(0,π]，我们构造一个函数φ，其径向延拓属于C_c^∞(ℝ^d)且支撑半径恰好为R，使得φ(‖x−y‖₂)在ℝ^d上严格正定，而φ(ρ(x,y))在𝕊^d上不正定。因此，每个可容许的支撑半径都可由一个光滑、严格欧氏正定的反例实现。
英文摘要
For every odd integer $d\geq3$, a continuous function $φ\colon[0,\infty)\to\mathbb R$ supported in $[0,π]$ 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 $R\in(0,π]$, we construct a function $φ$ whose radial extension belongs to $C_c^\infty(\mathbb R^d)$ and has support radius exactly $R$, such that $φ(\|\mathbf{x}-\mathbf{y}\|_2)$ is strictly positive definite on $\mathbb R^d$, whereas $φ(ρ(\mathbf{x},\mathbf{y}))$ is not positive definite on $\mathbb S^d$. Thus every admissible support radius is attained by a smooth, strictly Euclidean positive-definite counterexample.
Comments14 pages; minor expository revisions and clarifications, updated MSC classification, and added an acknowledgements section; results unchanged
