Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

TL;DR

This paper extends Schoenberg's classical theory to partially defined functions on discrete subsets of homogeneous spaces, providing new tools for constrained packing problems and code bounds.

Contribution

It develops a framework for partially defined complete positivity preservers on homogeneous spaces, generalizing classical results and applying them to packing and coding theory.

Findings

Characterization of partially defined completely positive functions on homogeneous spaces.

Application of the theory to derive bounds in packing problems.

Proof that polynomial functions witness sharpness in constrained angle codes.

Abstract

We develop a theory of partially defined complete positivity preservers, extending Schoenberg's classical characterization to functions defined only on discrete subsets or constrained domains. We frame the extension problem through the theory of completely positive maps on operator systems -- we characterize general partially defined completely positive definite functions on general homogeneous spaces. We apply our interpolation to constrained packing problems and Delsarte theory, where one uses positive definite functions on homogeneous spaces to obtain bounds on various packing problems. We prove the specific positive definite function witnesses that a code is sharp for constrained angle codes must be from polynomials.