Multivariate transforms of total positivity

TL;DR

This paper extends the classification of operators preserving total positivity from kernels to multivariate transforms, revealing that preservers are products of power functions in each variable for low orders and powers in a single variable for higher orders.

Contribution

It generalizes the classification of TP/TN preservers to multivariate transforms and higher orders, providing explicit characterizations and new insights into their structure.

Findings

Preservers of TP/TN of order 2 are products of individual power or Heaviside functions.

For higher orders, preservers are powers in a single variable.

Classifies multivariate transforms of symmetric TP/TN kernels, identifying specific product forms.

Abstract

Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified the post-composition operators that preserve TP/TN kernels of each specified order. We explain how to extend this from preservers to transforms, and from one to several variables. Namely, given arbitrary nonempty totally ordered sets $X,Y$, we characterize the transforms that send each tuple of kernels on $X \times Y$ that are TP/TN of orders $k_1, \dots, k_p$, to a TP/TN kernel of order $l$, for arbitrary positive integers (or infinite) $k_j$ and $l$. An interesting feature is that to preserve TP (or TN) of order $2$, the preservers are products of individual power (or Heaviside) functions in each variable; but for all higher orders, the preservers are powers in a single variable. We also classify the multivariate transforms of symmetric TP/TN kernels; in this case it is the preservers of TP/TN of order 3 that are multivariate products of power functions, and of order 4 that are individual powers. The proofs use generalized Vandermonde kernels, Hankel kernels, (strictly totally positive) Polya frequency functions, and a kernel studied recently but tracing back to works of Schoenberg [Ann. of Math. 1955] and Karlin [Trans. Amer. Math. Soc. 1964].