AI 中文总结
该研究证明偶维球面上各向同性正定函数可微性界的最优性，构造了满足特定可微性条件的函数并推广至所有偶维数，补充了此前仅奇维数已知的结论。
AI 中文摘要
我们证明了偶维球面上各向同性正定函数的可微性界的最优性。若此类函数在d维球面上的偶延拓在零处2k次可微，则该函数具有2k+⌊(d-1)/2⌋个连续内导数；此前最优性仅在奇维数中已知。我们在二维球面上构造了一个在赤道处一阶导数不存在的函数，并通过turning bands和spherical montée将其推广到所有偶维数。所得例子严格正定，在零处具有2k次可微性但不具备2k+2次，且在下一维数中不正定。
英文摘要
We prove optimality of the differentiability bound for isotropic positive definite functions on every even-dimensional sphere. If the even continuation of such a function on the $d$-dimensional sphere is $2k$ times differentiable at zero, then the function has $2k+\lfloor(d-1)/2\rfloor$ continuous interior derivatives; previously, optimality was known only in odd dimensions. We construct a function on the two-dimensional sphere whose first derivative does not exist at the equator and transfer it to all even dimensions by turning bands and spherical montée. The resulting examples are strictly positive definite, have $2k$ but not $2k+2$ derivatives at zero, and are not positive definite in the next dimension.
Comments15 pages
