There are many ways to upsample functions from multivariate scattered data locally, using only a few neighbouring data points of the evaluation point. The position and number of the actually used data points is not trivial, and many cases like Moving Least Squares require point selections that guarantee local recovery of polynomials up to a specified order. This paper suggests a kernel-based greedy local algorithm for point selection that has no such constraints. It realizes the optimal $L_\infty$ convergence rates in Sobolev spaces using the minimal number of points necessary for that purpose. On the downside, it does not care for smoothness, relying on fast $L_\infty$ convergence to a smooth function. The algorithm ignores near-duplicate points automatically and works for quite irregularly distributed point sets by proper selection of points. Its computational complexity is constant for each evaluation point, being dependent only on the Sobolev space parameters. Various numerical examples are provided. As a byproduct, it turns out that the well-known instability of global kernel-based interpolation in the standard basis of kernel translates arises already locally, independent of global kernel matrices and small separation distances.

翻译：从多元散乱数据中局部上采样函数的方法众多，这些方法仅使用评估点附近的少量邻域数据点。实际所用数据点的位置和数量并非无关紧要，许多方法（如移动最小二乘法）要求点选择必须保证局部重构达到指定阶数的多项式。本文提出一种基于核函数的贪心局部点选择算法，该算法不受此类约束限制。它利用实现该目的所需的最少点数，在Sobolev空间中实现了最优的$L_\infty$收敛速率。其不足之处在于不关注光滑性，仅依赖快速$L_\infty$收敛至光滑函数。该算法能自动忽略近重复点，并通过合理选择点集适用于分布极不规则的点集。每个评估点的计算复杂度为常数，仅依赖于Sobolev空间参数。文中提供了多种数值算例。作为副产品，研究发现基于核函数的全局插值在核平移标准基中众所周知的不稳定性其实在局部就已出现，这与全局核矩阵及小间距无关。