Intertwining of maxima of sum of translates functions with nonsingular kernels

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.