球上带限谱重构的正性损失
Positivity loss in bandlimited spectral reproduction on spheres
- School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对球上带限谱算子，证明其正性损失的精确阶为$(L/(N+1))^2$，并在圆、滤波超插值等场景阐释结果，识别出算子层面阻碍最大值原理的因素。
AI中文摘要：
对于能精确重构所有次数不超过$L$的模式的$N$阶带限谱算子，其正性损失能有多小？针对球面$\boldsymbol{\rm S}^d$上的球面多项式逼近，我们证明当$1\boldsymbol{\rm ≤}L<N$时，均匀算子范数超出1的最小超额（即最小正性损失）具有精确阶$\boldsymbol{\rm (L/(N+1))^2}$。下界由Fejér峰值检验和带限核的集中估计推导得出，而上界则通过用光滑滤波器校正正Jackson算子得到，二者匹配。我们在三种场景中阐释该结果：在圆上取$N=sL-1$时，这确定了广义投影常数超出1的精确阶，并明确了延迟de la Vallée--Poussin均值的$s^{-1}$超额与最优$s^{-2}$阶之间的差距；对于算子范数长期已知为一致有界的滤波超插值，我们给出其与正性阈值1之间差距的定量下界；最后，我们识别出算子层面上对最大值原理的阻碍。
英文摘要：
How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée--Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.
