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Characterization of the reproducing structure of the Bessel potential spaces beyond $p=2$

Tjeerd Jan Heeringa

Abstract

Reproducing kernel Hilbert spaces are uniquely characterized by their kernel, but reproducing kernel Banach spaces (RKBS) are not. However, a characterization of which RKBS admit a given kernel as reproducing kernel is lacking. This work provides such a characterization for the well-known Bessel potential / Matèrn kernel, a widely used covariance kernel for Gaussian processes which is the reproducing kernel of the Bessel potential space $H^{s,2}(\mathbb{R}^d)$ when $s>d/2$. Concretely, this work characterizes the pairs of Bessel potential spaces $H^{u,p}(\mathbb{R}^d),H^{v,q}(\mathbb{R}^d)$ which have this kernel.

Characterization of the reproducing structure of the Bessel potential spaces beyond $p=2$

Abstract

Reproducing kernel Hilbert spaces are uniquely characterized by their kernel, but reproducing kernel Banach spaces (RKBS) are not. However, a characterization of which RKBS admit a given kernel as reproducing kernel is lacking. This work provides such a characterization for the well-known Bessel potential / Matèrn kernel, a widely used covariance kernel for Gaussian processes which is the reproducing kernel of the Bessel potential space when . Concretely, this work characterizes the pairs of Bessel potential spaces which have this kernel.

Paper Structure

This paper contains 17 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Structure of the work
  5. Reproducing kernel Banach spaces
  6. Bessel potential spaces
  7. Proof of main results
  8. Discussion
  9. Lower bound for $s$
  10. Non-uniqueness of the kernel for a given pair
  11. Non-Hölder pairs
  12. Non-Hilbertian self-pairs
  13. Norming dual pairs
  14. Vanishing at infinity for $\max(p,q)=\infty$
  15. Strict inequality for $\max(p,q)<\infty$ and $\min(p,q)=1$
  16. ...and 2 more sections

Key Result

Theorem 1

Let $d\in \mathbb{N}$, $u,v,s>0$ and $1\leq p,q\leq \infty$. The spaces $H^{u,p}(\mathbb{R}^d),H^{v,q}(\mathbb{R}^d)$ are an RKBS pair with kernel $K_s$ if and only if with the inequality in eq:main_u+v_result being strict when $\min(p,q)=1$ and $\max(p,q)<\infty$, where $p'$ and $q'$ are the Hölder-conjugates of $p$ and $q$.

Theorems & Definitions (18)

  • Theorem 1
  • Definition 1: Reproducing kernel Banach space
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • ...and 8 more