Reproducing kernels and choices of associated feature spaces, in the form of $L^{2}$-spaces

TL;DR

This paper explores the relationship between positive definite kernels, their RKHS, and associated $L^{2}$-space feature representations, focusing on how to choose appropriate measures for specific applications in stochastic processes.

Contribution

It introduces a new analysis framework for kernels and their feature spaces in the form of $L^{2}$-spaces, tailored to applications in stochastic processes.

Findings

Characterization of measures suitable for specific kernels

Analysis of kernel and measure compatibility in $L^{2}$-spaces

Application-focused insights into kernel measure selection

Abstract

Motivated by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels $K$, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in the form $L^{2}\left(\mu\right)$; and we study the question of which measures $\mu$ are right for a particular kernel $K$. The answer to this depends on the particular application at hand. Such applications are the focus of the separate sections in the paper.