Minimal-Norm Extensions of Stationary Kernels on Subgroups of Locally Compact Abelian Groups and Gaussian Conditioning

Abstract

We study restrictions of stationary kernels on locally compact abelian groups $G$ to closed subgroups $H$. For a nonnegative spectral density $\hat{k}$, we derive an explicit fibrewise Fourier representation of the minimal-norm extension operator from the reproducing kernel Hilbert space of the restricted kernel on $H$ to the original space on $G$. We characterize when the canonical Fourier formula extends boundedly from $L^2(H)$ to $L^2(G)$, identify its exact operator norm and lower norm, and obtain bounds on the associated interpolation spaces. When $G$ is compact, the extension is a contraction and, for stationary Gaussian random variables admitting a measurable continuous version, maps the observed restriction to the conditional expectation. We also give a counterexample to a previously asserted supremum-norm contraction and illustrate the theory through cardinal interpolation and conditioning on one-dimensional subgroups of the torus.