逼近的映射视角：重缩放、变尺度与有理核方法的统一观点

中文摘要 AI 辅助

经典逼近方法通常通过改变采样集、增加数据点数量或选择不同的基函数来改进。我们提倡一种互补的观点：保持采样值不变，通过适当的映射来修改表示。这一观点统一了最初为不同目的而引入的若干构造。重缩放径向基函数插值通过一种强制精确再现常数的归一化来映射插值算子。变尺度核通过将数据提升到由尺度函数确定的高维流形来映射几何结构。映射基和伪节点在不重采样数据的情况下映射逼近空间，而有理核展开可解释为试验空间的非线性映射。我们为这些机制开发了统一的记号，总结了它们的逼近和稳定性性质，并解释了不连续变体与有理变体如何适配同一框架。可复现的数值实验展示了常数再现、通过数据模仿尺度函数实现的误差降低，以及映射多项式基对龙格现象的抑制。结果支持一个一般原则：映射提供了一种灵活的方式，在不修改原始观测的情况下，使逼近空间适应数据几何和正则性。

英文摘要

Classical approximation methods are usually improved by changing the sampling set, increasing the number of data points, or selecting a different basis. We advocate a complementary viewpoint: keep the sampled values fixed and modify the representation through a suitable mapping. This viewpoint unifies several constructions that were originally introduced for different purposes. Rescaled radial basis function interpolation maps the interpolation operator through a normalization that enforces exact reproduction of constants. Variably scaled kernels map the geometry by lifting the data to a higherdimensional manifold determined by a scale function. Mapped bases and fake nodes map the approximation space without resampling the data, while rational kernel expansions can be interpreted as nonlinear mappings of the trial space. We develop a common notation for these mechanisms, summarize their approximation and stability properties, and explain how discontinuous and rational variants fit the same framework. Reproducible numerical experiments illustrate constant reproduction, error reduction through data-mimicking scale functions, and the suppression of the Runge phenomenon by mapped polynomial bases. The results support a general principle: mappings provide a flexible way to adapt approximation spaces to data geometry and regularity without modifying the original observations

发表机构

• University of Padova（帕多瓦大学）
• Selcuk University（塞尔丘克大学）

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