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An algebra structure for reproducing kernel Hilbert spaces

Dimitrios Giannakis, Michael Montgomery

Abstract

Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions where pointwise evaluation is continuous. There are known examples of RKHSs that are Banach algebras under pointwise multiplication. These examples are built from weights on the dual of a locally compact abelian group. In this paper we define an algebra structure on an RKHS that is equivalent to subconvolutivity of the weight for known examples (referred to as reproducing kernel Hilbert algebras, or RKHAs). We show that the class of RKHAs is closed under the Hilbert space tensor product and the pullback construction on the category of RKHSs. The subcategory of RKHAs becomes a monoidal category with the spectrum as a monoidal functor to the category of topological spaces. The image of this functor is shown to contain all compact subspaces of $\mathbb R^n$ for $n>0$.

An algebra structure for reproducing kernel Hilbert spaces

Abstract

Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions where pointwise evaluation is continuous. There are known examples of RKHSs that are Banach algebras under pointwise multiplication. These examples are built from weights on the dual of a locally compact abelian group. In this paper we define an algebra structure on an RKHS that is equivalent to subconvolutivity of the weight for known examples (referred to as reproducing kernel Hilbert algebras, or RKHAs). We show that the class of RKHAs is closed under the Hilbert space tensor product and the pullback construction on the category of RKHSs. The subcategory of RKHAs becomes a monoidal category with the spectrum as a monoidal functor to the category of topological spaces. The image of this functor is shown to contain all compact subspaces of for .
Paper Structure (6 sections, 14 theorems, 79 equations)

This paper contains 6 sections, 14 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Results from RKHS theory
  3. Reproducing kernel Hilbert algebras
  4. The RKHA category
  5. The GRS condition
  6. Banach algebra quotients of RKHAs

Key Result

Theorem 1.2

Aronszajn50 Let $X$ be a set and $k$ a positive definite kernel on $X$. Then there exists a unique reproducing kernel Hilbert space $\mathcal{H} \subset \mathcal{L}(X)$ with $k$ as its kernel.

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of theorem
  • ...and 30 more