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

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.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-11-11

• Fields of study

  Mathematics

• Identifiers
  arXiv 2511.08406
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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