Total Positivity of Analytic Bases through Symmetric Functions

TL;DR

This paper explores the total positivity of analytic bases' collocation matrices via symmetric functions, providing explicit formulas, conditions for positivity, and generalizations of classical identities.

Contribution

It introduces explicit formulas for minors, establishes conditions for total positivity, and generalizes the Cauchy identity using symmetric functions.

Findings

Derived explicit formulas for initial minors in terms of Schur functions

Established sufficient conditions for total positivity of analytic systems

Generalized the Cauchy identity for specific function families

Abstract

This paper studies the bidiagonal factorization of the collocation matrices of analytic bases using symmetric functions. Explicit formulas for their initial minors are derived in terms of Schur functions. The structure of these formulas permits establishing sufficient conditions for the total positivity of generic systems of analytic functions. In addition, they have been found to lead to generalizations of the Cauchy identity for certain families of functions.