The Positive-Definite Completion Problem

TL;DR

This paper investigates the positive-definite completion problem for kernels, establishing existence, uniqueness, and characterizations of solutions, including a canonical completion analogous to determinant-maximizing matrix completions.

Contribution

It introduces the concept of a canonical completion for kernels, providing algebraic and variational characterizations and proving its existence and uniqueness under specific domain conditions.

Findings

Existence of canonical completion for domains containing the diagonal band

Uniqueness of the canonical completion under certain conditions

Characterizations of solutions via algebraic and variational methods

Abstract

We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band containing the diagonal. This corresponds to the existence of a canonical extension in the context of the classical extension problem of positive-definite functions, which can be understood as the solution to an abstract Cauchy problem in a certain reproducing kernel Hilbert space.