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On the extension of positive definite kernels to topological algebras

Daniel Alpay, Ismael L. Paiva

Abstract

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras, and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

On the extension of positive definite kernels to topological algebras

Abstract

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras, and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

Paper Structure

This paper contains 5 sections, 11 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Extension of kernels to topological algebras
  3. Analytic kernels for entire functions
  4. General analytic kernels
  5. Operators

Key Result

Proposition 2.2

The factorization holds and the reproducing kernel Hilbert space associated to Eq. missouri consists of functions of the form with inner product and norm induced from the inner product and the norm of $\mathcal{G}$.

Theorems & Definitions (20)

  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 10 more