Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods
Abstract
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
Related
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BibTeX
@misc{marchi2026mapping,
title = {{Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods}},
author = {Stefano De Marchi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07743},
eprint = {2609.07743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07743},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mapping for Approximation: A Unified View of Rescaled, Variably Scaled and Rational Kernel Methods
AU - Stefano De Marchi
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07743
UR - https://arxiv.org/abs/2609.07743
AB - 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
ER -