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General superconvergence for kernel-based approximation

Toni Karvonen, Gabriele Santin, Tizian Wenzel

TL;DR

This work addresses superconvergence in kernel-based approximation by developing a general operator-range framework that identifies when smoother target functions yield accelerated convergence. It establishes a main result: if the target lies in the range of the adjoint of a linear operator, then projection errors improve by a factor tied to the operator-adjoint norm, with a continuum of intermediate rates via real interpolation spaces. Specialization to Mercer-based spaces yields power spaces $\mathcal{H}_\theta(\Omega)$, linking fractional smoothness to convergence via the Mercer spectrum, and showing that $T(L_2(\Omega))$ plays a central role in fractional rates. The paper also extends to general kernel integral operators $A^*$ and characterizes images like $T(L_p(\Omega))$, revealing dense but non-surjective embeddings and the critical influence of boundary conditions in Sobolev settings. Numerical experiments on Sobolev, Wendland, and periodic kernels corroborate the theory and illustrate how boundary constraints and intermediate smoothness affect achievable rates in practice.

Abstract

Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting, research has focused on two regimes: misspecified interpolation, where the kernel smoothness exceeds that of the target function, and superconvergence, where the target is smoother than the Hilbert space. This work addresses the latter, where smoother target functions yield improved convergence rates, and extends existing results by characterizing superconvergence for projections in general Hilbert spaces. We show that functions lying in ranges of certain operators, including adjoint of embeddings, exhibit accelerated convergence, which we extend across interpolation scales between these ranges and the full Hilbert space. In particular, we analyze Mercer operators and embeddings into $L_p$ spaces, linking the images of adjoint operators to Mercer power spaces. Applications to Sobolev spaces are discussed in detail, highlighting how superconvergence depends critically on boundary conditions. Our findings generalize and refine previous results, offering a broader framework for understanding and exploiting superconvergence. The results are supported by numerical experiments.

General superconvergence for kernel-based approximation

TL;DR

This work addresses superconvergence in kernel-based approximation by developing a general operator-range framework that identifies when smoother target functions yield accelerated convergence. It establishes a main result: if the target lies in the range of the adjoint of a linear operator, then projection errors improve by a factor tied to the operator-adjoint norm, with a continuum of intermediate rates via real interpolation spaces. Specialization to Mercer-based spaces yields power spaces , linking fractional smoothness to convergence via the Mercer spectrum, and showing that plays a central role in fractional rates. The paper also extends to general kernel integral operators and characterizes images like , revealing dense but non-surjective embeddings and the critical influence of boundary conditions in Sobolev settings. Numerical experiments on Sobolev, Wendland, and periodic kernels corroborate the theory and illustrate how boundary constraints and intermediate smoothness affect achievable rates in practice.

Abstract

Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting, research has focused on two regimes: misspecified interpolation, where the kernel smoothness exceeds that of the target function, and superconvergence, where the target is smoother than the Hilbert space. This work addresses the latter, where smoother target functions yield improved convergence rates, and extends existing results by characterizing superconvergence for projections in general Hilbert spaces. We show that functions lying in ranges of certain operators, including adjoint of embeddings, exhibit accelerated convergence, which we extend across interpolation scales between these ranges and the full Hilbert space. In particular, we analyze Mercer operators and embeddings into spaces, linking the images of adjoint operators to Mercer power spaces. Applications to Sobolev spaces are discussed in detail, highlighting how superconvergence depends critically on boundary conditions. Our findings generalize and refine previous results, offering a broader framework for understanding and exploiting superconvergence. The results are supported by numerical experiments.
Paper Structure (22 sections, 15 theorems, 108 equations, 6 figures)

This paper contains 22 sections, 15 theorems, 108 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. RKHS and orthogonal projections
  4. Mercer expansions
  5. Sobolev spaces and kernels
  6. Superconvergence
  7. General superconvergence
  8. Intermediate orders
  9. Superconvergence in Mercer-based spaces
  10. Power spaces and interpolation
  11. Fractional superconvergence
  12. Superconvergence in the image of general kernel integral operators
  13. General kernel integral operators
  14. Characterization of images
  15. Boundary conditions
  16. ...and 7 more sections

Key Result

Theorem 3

Let $\Omega\subset\mathbb{R}^d$ be a bounded set with a Lipschitz boundary that satisfies an interior cone condition. Let $q\in[1,\infty]$, $m\in\mathbb{N}_0$, and let $k$ be a Sobolev kernel of order $\tau>m+d/2$. Then there are constants $C, h_0 >0$ such that for all discrete sets $X\subset\Omega$ where $(x)_+\coloneqq \max(0,x)$.

Figures (6)

  • Figure 1: Visualization of the scale of power spaces with the special cases $L_2(\Omega)$ (for $\theta = 0$), $\mathcal{H}(\Omega)$ (for $\theta = 1$) and $TL_2(\Omega)$ (for $\theta = 2$). In the case of Sobolev kernels, $TL_1(\Omega)$ refers to the power space with index $2 - \frac{d}{2\tau}$, see \ref{['th:embedding_power_spaces']}. This paper considers the superconvergence case of intermediate order, i.e. $\theta \in (1, 2]$.
  • Figure 2: Visualization of the convergence rates for interpolation error of $f_\alpha$ ($y$-axis) in the $L_1(\Omega)$, $L_2(\Omega)$, $L_\infty(\Omega)$ norm in terms of the fill distance $h_{X}$ over the parameter $\alpha$ ($x$-axis) for $\Omega = [0, 1]^d$, $d=1$ (left) and $d=2$ (right). From top to bottom: Basic, linear and quadratic Matérn kernel (see \ref{['eq:matern_kernels']}).
  • Figure 3: Visualization of the convergence rates for interpolation error of $f_\alpha$ ($y$-axis) in the $L_1(\Omega), L_2(\Omega), L_\infty(\Omega)$ norms in terms of the fill distance $h_{X}$ over the parameter $\alpha$ ($x$-axis) for $\Omega = [0, 1]$ and a Wendland kernel of low smoothness.
  • Figure 4: Convergence rates of the interpolation error of $f_{i,\alpha}$ for $i=1, 2, 3$, plotted as functions of $\alpha\in(3/2, 6]$, for the $L_\infty$ norm (left) and the $L_2$ norm (right). The horizontal dotted lines report the observed saturating values, while the theoretically expected rates are marked with crosses ($\mathcal{H}(\Omega)$ and $T(L_2(\Omega))$), and with diagonal dashed lines (for the intermediate spaces).
  • Figure 5: Five functions $f_\alpha$ on $[0, 1]$ sampled according to \ref{['eq:periodic-sample-funcs']} for each $\alpha \in \{0.3, 0.8, 1.3\}$.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Remark 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • proof
  • Remark 7
  • Theorem 8
  • proof
  • Remark 9
  • ...and 30 more