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Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

Sujit Sakharam Damase, James Eldred Pascoe

TL;DR

This work extends Schoenberg's classical theory of positivity preservers to partially defined and operator-valued settings on homogeneous spaces, using completely positive maps and operator-system techniques. It develops a discrete, finite-set framework (including the Hamming cube) and ties it to continuous, invariant CPD functions on groups and homogeneous spaces, enabling interpolation and extension results. Key contributions include a CPD-based discrete Schoenberg theorem, a representation-theoretic treatment of invariant CPD functions, a partial characterization of partially defined complete positivity preservers, and applications to constrained angle codes and Delsarte-type bounds. The results unify harmonic analysis, operator algebras, and combinatorial coding theory to derive sharp bounds and structure results for packing problems, with implications for classical bounds (Delsarte, Lovász theta) and modern asymptotic frameworks (Olshanski spherical pairs).

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.

Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

TL;DR

This work extends Schoenberg's classical theory of positivity preservers to partially defined and operator-valued settings on homogeneous spaces, using completely positive maps and operator-system techniques. It develops a discrete, finite-set framework (including the Hamming cube) and ties it to continuous, invariant CPD functions on groups and homogeneous spaces, enabling interpolation and extension results. Key contributions include a CPD-based discrete Schoenberg theorem, a representation-theoretic treatment of invariant CPD functions, a partial characterization of partially defined complete positivity preservers, and applications to constrained angle codes and Delsarte-type bounds. The results unify harmonic analysis, operator algebras, and combinatorial coding theory to derive sharp bounds and structure results for packing problems, with implications for classical bounds (Delsarte, Lovász theta) and modern asymptotic frameworks (Olshanski spherical pairs).

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.
Paper Structure (47 sections, 52 theorems, 109 equations)

This paper contains 47 sections, 52 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Motivation
  4. Statistics and numerics
  5. Floating point arithmetic
  6. Conservative interpolation
  7. Nevanlinna-Pick interpolation
  8. Sphere packing and Schoenberg's theorem
  9. Lovàsz theta function and Schoenberg's theorem
  10. Cone preservers
  11. Group representation theory and positive definite functions on groups
  12. Homogeneous spaces
  13. Group representation theory
  14. Representation theory of compact groups
  15. Ultraspherical polynomials
  16. ...and 32 more sections

Key Result

Theorem 1.1

If $\psi: \mathbb{Z} \rightarrow \mathbb{R}$ is a partially defined positivity preserver then where $a_n\geq 0$ and $\limsup a_n^{1/n} =0.$

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 2.1: Riesz representation theorem
  • Definition 2.2: Reproducing kernel Hilbert spaces
  • Theorem 2.3: Aronszajn-Moore-Kolmogorov type theorem
  • Theorem 2.4: Pick pick, Nevanlinna nevanlinna
  • Theorem 2.5: Delsarte–-Goethals–-Seidel DGS, Kabatiansky–-Levenshtein K-L, Cohn--Zhao Cohn-Zhao
  • proof
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 80 more