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Which Spaces can be Embedded in Reproducing Kernel Hilbert Spaces?

Max Schölpple, Ingo Steinwart

Abstract

Given a Banach space $E$ consisting of functions, we ask whether there exists a reproducing kernel Hilbert space $H$ with bounded kernel such that $E\subset H$. More generally, we consider the question, whether for a given Banach space consisting of functions $F$ with $E\subset F$, there exists an intermediate reproducing kernel Hilbert space $E\subset H\subset F$. We provide both sufficient and necessary conditions for this to hold. Moreover, we show that for typical classes of function spaces described by smoothness there is a strong dependence on the underlying dimension: the smoothness $s$ required for the space $E$ needs to grow \emph{proportional} to the dimension $d$ in order to allow for an intermediate reproducing kernel Hilbert space $H$.

Which Spaces can be Embedded in Reproducing Kernel Hilbert Spaces?

Abstract

Given a Banach space consisting of functions, we ask whether there exists a reproducing kernel Hilbert space with bounded kernel such that . More generally, we consider the question, whether for a given Banach space consisting of functions with , there exists an intermediate reproducing kernel Hilbert space . We provide both sufficient and necessary conditions for this to hold. Moreover, we show that for typical classes of function spaces described by smoothness there is a strong dependence on the underlying dimension: the smoothness required for the space needs to grow \emph{proportional} to the dimension in order to allow for an intermediate reproducing kernel Hilbert space .
Paper Structure (16 sections, 29 theorems, 174 equations)

This paper contains 16 sections, 29 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Results
  4. Applications to Various Concrete Spaces
  5. Warm-up: LG
  6. Spaces of Hölder Continuous Functions
  7. Spaces of (Generalized) Mixed Smoothness
  8. Embedding Integral Reproducing Kernel Banach Spaces into RKHSs
  9. Triebel-Besov-Lizorkin spaces
  10. Proofs
  11. Proofs for Section \ref{['sec:general']}
  12. Proofs for \ref{['sec:examples']}
  13. Proofs for \ref{['subsection:abstact_holder']}
  14. Proofs for \ref{['subsubsection:sobolev_spaces_of_mixed_smoothness']}
  15. Proofs for
  16. ...and 1 more sections

Key Result

Proposition 2

Let $F$ be a proper BSF on $X$ and $E$ be a closed subspace of $F$. Then $E$ is a proper BSF and the following statements are equivalent: Moreover, in this case we have $E\hookrightarrow H \hookrightarrow F$.

Theorems & Definitions (53)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Definition 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Corollary 9
  • Lemma 10
  • ...and 43 more