AI 中文总结
该研究针对球面上的各向同性正定函数，证明奇维数下已知的正定性结论在偶维数下不成立，还给出二维显式反例，解决了偶维Gneiting问题。
AI 中文摘要
我们研究欧氏空间上紧支撑各向同性正定函数在将欧氏距离替换为球面测地距离后是否仍保持正定性。该结论在奇维数下已知成立。我们证明在任意偶维数下该结论不成立，无论对支撑集施加多小的正上界，还给出了二维情况下的显式反例。
英文摘要
We study when a compactly supported isotropic positive definite function on a Euclidean space remains positive definite after Euclidean distance is replaced by spherical geodesic distance. The corresponding assertion is known in odd dimensions. We prove that it fails in every even dimension, no matter how small a positive upper bound is imposed on the support. An explicit counterexample is also given in dimension two.
