Intertwining of maxima of sum of translates functions with nonsingular kernels

TL;DR

This paper studies the maxima of sum of translates functions with singular and nonsingular kernels, revealing an intertwining property of local maxima across different node configurations, extending previous results.

Contribution

It extends the analysis of maxima of sum of translates functions to include nonsingular kernels, generalizing the intertwining property previously known for singular kernels.

Findings

Established the intertwining property for functions with singular kernels.

Partially extended the intertwining property to nonsingular kernels.

Analyzed the behavior of local maxima in sum of translates functions.

Abstract

In previous papers we investigated so-called sum of translates functions $F({\mathbf{x}},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb{R}}:={\mathbb{R}}\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb{R}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0} K(t)=-\infty$. For node systems ${\mathbf{x}}:=(x_1,\ldots,x_n)$ with $x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector ${\mathbf{m}}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j({\mathbf{x}}):=\sup_{x_j\le t\le x_{j+1}} F({\mathbf{x}},t)$. Among other results we proved a strong intertwining property: if the kernels are also decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i({\mathbf{x}})<m_i({\mathbf{y}})$ and $m_j({\mathbf{x}})>m_j({\mathbf{y}})$. Here we partially succeed to extend this even to nonsingular kernels.