Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

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.

💡 Research Summary

The paper develops a comprehensive theory that extends Schoenberg’s classical characterization of positivity preservers to functions that are only defined on discrete subsets or constrained domains, and further lifts the theory to the setting of completely positive (CP) maps on operator systems. The authors begin by introducing the notion of a partially defined positive semidefinite (PSD) matrix, where some entries are left unspecified. A matrix is called PSD‑completable if the unspecified entries can be filled in so that the whole matrix becomes PSD. This concept mirrors practical situations such as covariance matrix completion with missing data.

With this language, a “partial positivity preserver” f : X→ℝ (X⊂