发表机构
School of Mathematics, Guangxi University（广西大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究积分几何中方向平均的有限精确替代，通过范数恒等式和谱判据给出核自适应设计，并分类了幂核的精确性，对柯西核给出误差界。
AI 中文摘要
我们研究积分几何中的不变方向平均何时可以被有限多个方向精确替代。对于偶连续核 $\psi:[-1,1]\to\R$，设 $T_\psi$ 为球面上相应的带状卷积算子，并设 $\cB_\psi K$ 为凸体 $K\subset\R^d$ 的相应表面积观测量。我们的第一个结果是尖锐的范数恒等式：对所有凸体而言，最坏相对误差等于势差 $T_\psi(\mu-\sigma)$ 的 $L^\infty$ 范数的一半，其中 $\mu$ 为任意归一化带符号方向测度。因此，普适精确性等价于 $\mu-\sigma\in\ker T_\psi$。Funk-Hecke 对角化随后给出完整的谱判据：必须恰好消去 $T_\psi$ 的乘子非零的那些球谐次数。这产生了核自适应设计以及有限活动谱的正有限精确规则。对于 $\psi_p(s)=|s|^p$，我们得到完全分类：偶次幂精确给出加权实射影设计，而非偶次幂则不存在对每个凸体都精确的有限带符号原子规则。对于经典柯西核 $|s|$，精确性失败，但球面 $t$-设计给出均匀的 $O(t^{-1})$ 相对表面积误差。我们还推导了可求长子流形的精确投影矩恒等式。
英文摘要
We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel $ψ:[-1,1]\to\R$, let $T_ψ$ be the associated zonal convolution operator on the sphere and let $\cB_ψK$ be the corresponding surface-area observable of a convex body $K\subset\R^d$. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the $L^\infty$ norm of the potential discrepancy $T_ψ(μ-σ)$, for every normalized signed directional measure $μ$. Hence universal exactness is equivalent to $μ-σ\in\ker T_ψ$. Funk--Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of $T_ψ$ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For $ψ_p(s)=|s|^p$ we obtain a complete classification: even powers give precisely weighted real projective designs, whereas non-even powers admit no finite signed atomic rule that is exact for every convex body. For the classical Cauchy kernel $|s|$, exactness fails but spherical $t$-designs give a uniform $O(t^{-1})$ relative surface-area error. We also derive exact projection-moment identities for rectifiable submanifolds.
