Kernel approximation beyond the native space -- with applications to approximation on manifolds

2026/06/30 by Thomas Hangelbroek, Christian Rieger, Grady B. Wright
Mathematics · Computer Science · #math.CA #cs.NA #math.NA #msc:65D12 #msc:46E35 #msc:47B34 #msc:58J40

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arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

This article treats kernel approximation and interpolation on embedded manifolds of ℝ^Nusing restrictions of positive and conditionally positive definite kernels. The main challenge is to develop an approximation theory that treats error measured in highly regular smoothness spaces relative to the kernel. This means that the order of smoothness is higher than that of the kernel's associated native space (in the positive definite case, the reproducing kernel Hilbert space generated by the kernel). This prevents the use of standard techniques for controlling error in this setting, especially RKHS space arguments like orthogonality of the interpolation projector, or bounds using the \em power function. To address this challenge, we extend methods for treating target functions given as potentials introduced by DeVore and Ron, and give conditions guaranteeing that restrictions of such functions span Sobolev smoothness spaces on the manifolds. Together with kernel-based Bernstein inequalities for embedded manifolds, these results give a comprehensive theory for interpolation error. As an application of the theory, we derive new error estimates for approximating the eigenspace of certain differential operators arising from kernel-based methods for partial differential equations on manifolds.

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