Extension of positive definite functions

TL;DR

This paper establishes criteria for extending positive definite functions from a subset of Euclidean space to the whole space, linking extensions to unitary representations and spectral properties, with applications to stochastic processes.

Contribution

It provides necessary and sufficient conditions for extending positive definite functions, connecting these extensions to unitary representations and spectral analysis, and extends results to conditionally negative definite functions.

Findings

Conditions for extension involve strong commutativity of unbounded operators

Extensions correspond to different unitary representations with simple spectrum

Criteria for unique extension of positive definite functions

Abstract

Let $\Omega\subset\mathbb{R}^n$ be an open, connected subset of $\mathbb{R}^n$, and let $F\colon\Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of a system of certain unbounded selfadjoint operators defined on a Hilbert space associated to $F$. When a positive definite function $F$ is extendable, we show that it is characterized by existence of associated unitary representations of $\mathbb{R}^n$. Different positive definite extensions correspond to different unitary representations. We prove that each such unitary representation has simple spectrum. We give necessary and sufficient conditions for a continuous positive definite function to have exactly one extension. Our proof regarding extensions of positive definite functions carries over mutatis mutandis to the case of conditionally negative definite functions, which has applications to Gaussian stochastic processes, whose increments in mean-square are stationary (e.g., fractional Brownian motion).