The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

TL;DR

This survey explores functions that preserve positive semidefiniteness when applied entrywise across all dimensions, highlighting their mathematical properties, applications, and connections to sphere packings and metric geometry.

Contribution

It provides a comprehensive overview of dimension-free positivity preservers, including classical results, recent developments, and an appendix on sphere packings and lattice structures.

Findings

Classification of positive definite functions on spheres

Connections between positivity preservers and metric embeddings

Application of Delsarte's linear programming to sphere packings

Abstract

We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg, Rudin, and Loewner, the subject has attracted renewed attention owing to its connections to various applied fields and techniques. In this survey we will focus mainly on the question of preserving positivity in all dimensions. Connections to distance geometry and metric embeddings, positive definite sequences and functions, Fourier analysis, applications and covariance estimation, Schur polynomials, and finite fields will be discussed. The Appendix contains a mini-survey of sphere packings, kissing numbers, and their "lattice" versions. This part overlaps with the rest of the article via Schoenberg's classification of the positive definite functions on spheres, aka dimension-free entrywise positivity preservers with a rank constraint - applied via Delsarte's linear programming method.