Spectral Theory for Mercer Operators, and Implications for $Ext\left (F\right )$

Abstract

Given a continuous positive definite (p.d.) function F on the open interval $\left (-1,1\right )$, we are concerned with the set $Ext\left (F\right )$ of its extensions to p.d. functions defined on all of $\mathbb{R}$, as well as a certain subset $Ext_{1}\left (F\right )$ of $Ext\left (F\right )$. Since every such p.d. extension of F is a Bochner transform of a unique positive and finite Borel measure μ on $\mathbb{R}$, i.e., $\widehat{d\mu }\left (x\right )$, $x \in \mathbb{R}$ and $\mu \in \mathcal{M}_{+}\left (\mathbb{R}\right )$, we will speak of $Ext\left (F\right )$ as a subset of $\mathcal{M}_{+}\left (\mathbb{R}\right )$. The purpose of this chapter is to gain insight into the nature and properties of $Ext_{1}\left (F\right )$.