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

2026/08/05 by Daniel Winkle
Mathematics · #math.PR #msc:60G15 #msc:46E22

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arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We study restrictions of stationary kernels on locally compact abelian groups G to closed subgroups H. For a nonnegative spectral density 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 L²(H) to L²(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.

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