范围控制匹配和离散数据拟插值的等周组合界
Isoperimetric-Combinatorial Bounds for Range-Controlled Matchings and Quasi-Interpolation from Scattered Data
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中文总结 AI 辅助
该研究开发介观框架分析有限点集扰动，通过施加HT组合约束证明$Y$与$Q$存在范围为$O(r)$的完美匹配，可转移积分逼近估计并锚定$Q$，还用平移不变拟插值方法获高阶估计，关键限制是HT条件及$r\leq Ch$。
中文摘要 AI 辅助
我们开发了一个用于分析有限点集扰动的介观框架。给定具有已知求积和逼近性质的参考节点集$Y$，考虑仅通过其在尺度$r>0$的立方体中的总体来观测的无序节点集$Q$。通过对这些总体施加霍尔型（HT）组合约束，我们证明了$Y$和$Q$之间存在范围为$O(r)$的完美匹配。这使得在尺度$h>r$的较粗立方体上的积分逼近估计能够从$Y$转移到$Q$，并带有明确的误差控制，还将$Q$锚定到周期网格。然后我们使用平移不变拟插值方法，如同在拟均匀设置中一样获得$h^s$阶的高阶估计，但针对不同的几何类。关键限制是HT条件和$r\leq Ch$的界，其中$C<1$与尺度无关。
英文摘要
We develop a mesoscopic framework for analyzing perturbations of finite point sets. Given a reference node set $Y$ with known cubature and approximation properties, we consider a disordered node set $Q$ that is observed only through its populations in cubes at scale $r>0$. By imposing Hall-type (HT) combinatorial constraints on these populations, we prove the existence of a perfect matching between $Y$ and $Q$ with $O(r)$ range. This allows integral approximation estimates on coarser cubes at scale $h>r$ to be transferred from $Y$ to $Q$ with explicit error control and anchors $Q$ to a periodic grid. We then use translation-invariant quasi-interpolation methods to obtain high-order estimates of order $h^s$ as in the quasi-uniform setting, but for a different class of geometries. The key restrictions are the HT conditions and the bound $r\le Ch$, where $C<1$ is scale independent.
