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Total Positivity of Analytic Bases through Symmetric Functions

Pablo Díaz, Esmeralda Mainar

TL;DR

The paper addresses establishing total positivity criteria for collocation matrices of analytic bases by expressing their initial minors in terms of Schur polynomials and the infinite Wronskian $W_{\mathbf f}$. It extends previous polynomial-case results to analytic function bases, proving an explicit Schur-basis expansion for initial minors and deriving Cauchy-type identities as special cases. A key contribution is a practical sufficiency condition: if the infinite Wronskian $W_{\mathbf f}$ is TP, then the analytic basis is TP on $(0,\infty)$, and the framework recovers classical Cauchy identities as particular instances. The work provides a unified theoretical and computational toolkit for TP analysis with potential applications to Wronskian and Gram matrices and precision numerics in analytic settings.

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.

Total Positivity of Analytic Bases through Symmetric Functions

TL;DR

The paper addresses establishing total positivity criteria for collocation matrices of analytic bases by expressing their initial minors in terms of Schur polynomials and the infinite Wronskian . It extends previous polynomial-case results to analytic function bases, proving an explicit Schur-basis expansion for initial minors and deriving Cauchy-type identities as special cases. A key contribution is a practical sufficiency condition: if the infinite Wronskian is TP, then the analytic basis is TP on , and the framework recovers classical Cauchy identities as particular instances. The work provides a unified theoretical and computational toolkit for TP analysis with potential applications to Wronskian and Gram matrices and precision numerics in analytic settings.

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.
Paper Structure (5 sections, 7 theorems, 50 equations)

This paper contains 5 sections, 7 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Total positivity and symmetric functions
  3. Initial minors of collocation matrices and symmetric functions
  4. Explicit formulation of Cauchy-type identities
  5. Conclusions and further research

Key Result

theorem 1

Let $\mathbf{f} = (f_1, \ldots, f_n)$ be a system of analytic functions on $I \subseteq \mathbb{R}$ and $\mathbf{x} = \{ x_1 , \ldots, x_n \}$ in $I$ satisfying $x_1 < \cdots < x_n$. Then and where, $0 < j \leq i \leq n$, and, for any $\lambda=(\lambda_1,\ldots,\lambda_j) \in \Lambda$,

Theorems & Definitions (16)

  • theorem 1
  • proof
  • corollary thmcountercorollary
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • theorem 4
  • proof
  • ...and 6 more