[Paper Review] Characterizing matrix monotonicity of fixed order on general sets

Otte Heinävaara|arXiv (Cornell University)|Jun 14, 2019

Mathematical Inequalities and Applications16 references4 citations

TL;DR

This paper provides new characterizations of $n$-monotone and $n$-convex functions on general subsets of $\mathbb{R}$ using higher-order divided differences, unifying and extending classical results by Loewner, Dobsch, and Donoghue. The key contribution is a necessary and sufficient condition for $n$-monotonicity on arbitrary sets: non-negativity of divided differences of $fq^2$ of order $2k-1$ for all $1 \leq k \leq n$ and $q \in \mathbb{C}_{k-1}[x]$, with a sharp counterexample showing extension to convex hulls is not always possible.

ABSTRACT

We give new characterizations for matrix monotonicity and convexity of fixed order which connects previous characterizations by Loewner, Dobsch, Donoghue, Kraus and Bendat--Sherman. The ideas introduced are then used to characterize matrix monotone functions of arbitrary order on general subsets of the real line.

Motivation & Objective

• To unify and generalize classical characterizations of $n$-monotone and $n$-convex functions using higher-order divided differences.
• To extend the theory of $n$-monotonicity from intervals to arbitrary subsets of $\mathbb{R}$, addressing a gap in the literature.
• To establish that $n$-monotonicity is a local property on general sets, even when the domain is not an interval.
• To provide a sharp counterexample showing that $n$-monotone functions on finite sets cannot always be extended to their convex hulls.

Proposed method

• Introduce a new characterization of $n$-monotonicity using divided differences of order $2k-1$ for $fq^2$, where $f$ is defined on a set $F$ and $q$ is a polynomial of degree $k-1$.
• Use the Peano kernel representation of divided differences to connect the new characterization with existing matrix-based characterizations.
• Prove that $n$-monotonicity on a set $F$ is equivalent to the non-negativity of $[x_0, \dots, x_{2k-1}]_{fq^2} \geq 0$ for all $1 \leq k \leq n$ and all pairwise distinct $x_i \in F$, with $q \in \mathbb{C}_{k-1}[x]$.
• Construct a counterexample using rational functions $r_1$ and $r_2$ of degree $n-1$ that agree on $2n$ points but differ outside, showing that $f$ cannot be extended to a point in the convex hull without violating $n$-monotonicity.
• Leverage the fact that $n$-monotonicity implies $k$-tone for $k = 2n-1$, and use this to derive regularity and non-extendability results.
• Apply the characterization to prove that $2$-monotone functions on unbounded sets must be affine, using the behavior of triple divided differences under scaling.

Experimental results

Research questions

• RQ1 Can $n$-monotonicity on general subsets of $\mathbb{R}$ be characterized by conditions on divided differences of $fq^2$?
• RQ2 Is $n$-monotonicity a local property on arbitrary sets, even when the set is not an interval?
• RQ3 Can every $n$-monotone function on a finite set be extended to its convex hull while preserving $n$-monotonicity?
• RQ4 What is the precise regularity implied by $n$-monotonicity on general sets, and how does it relate to divided differences?
• RQ5 Under what conditions on the domain does $n$-monotonicity force a function to be affine?

Key findings

• A function $f:F\to\mathbb{R}$ is $n$-monotone if and only if $[x_0,\dots,x_{2k-1}]_{fq^2} \geq 0$ for all $1 \leq k \leq n$, all pairwise distinct $x_i \in F$, and all $q \in \mathbb{C}_{k-1}[x]$.
• If $\#F > 2n$, it suffices to verify the condition only for $k = n$, simplifying verification on large finite sets.
• The theory of $n$-monotonicity is local: if $f$ is $n$-monotone on overlapping intervals, it is $n$-monotone on their union, even when the domain is not an interval.
• There exists a finite set $F$ and a $n$-monotone function $f:F\to\mathbb{R}$ that cannot be extended to any point in $\operatorname{conv}(F)\setminus F$, showing that convex hull extension is not always possible.
• If $F$ is unbounded and $f:F\to\mathbb{R}$ is $2$-monotone, then $f$ must be affine, as shown by the vanishing of triple divided differences under scaling.
• The $n$-monotonicity of $f$ implies that $f$ is $(2n-1)$-tone, meaning all divided differences of order $2n-1$ are non-negative, which implies $f \in C^{2n-3}$.