The entrywise calculus and dimension-free positivity preservers, with an Appendix on 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.
Keywords
Cite
@article{arxiv.2511.08406,
title = {The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings},
author = {Apoorva Khare},
journal= {arXiv preprint arXiv:2511.08406},
year = {2026}
}
Comments
A self-contained Appendix on sphere packings and kissing numbers, follows the main survey on positivity preservers. (2 figures, 2 tables.) This survey was written on the occasion of the 90th anniversary of the Indian National Science Academy (INSA). This updated and final version has minor edits, e.g. to Theorem A.24. Also added a paragraph after Question 6.9, and added the reference [39]