Harmonic analysis of network systems via kernels and their boundary realizations

Abstract

<p style='text-indent:20px;'>With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) <inline-formula><tex-math id="M1">\begin{document}$ K $\end{document}</tex-math></inline-formula> and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel <inline-formula><tex-math id="M2">\begin{document}$ K $\end{document}</tex-math></inline-formula> we analyze associated Gaussian processes <inline-formula><tex-math id="M3">\begin{document}$ V $\end{document}</tex-math></inline-formula>. Properties of the Gaussian processes will be derived from certain factorizations of <inline-formula><tex-math id="M4">\begin{document}$ K $\end{document}</tex-math></inline-formula>, arising as a covariance kernel of <inline-formula><tex-math id="M5">\begin{document}$ V $\end{document}</tex-math></inline-formula>. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula>. Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loève expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.</p>