A homeomorphism theorem for sums of translates

TL;DR

This paper establishes a homeomorphism theorem for sums of translates of concave functions, analyzing the structure of maximum values over intervals as nodes vary, with applications in interpolation theory and approximation.

Contribution

It introduces a novel homeomorphism theorem for sums of translates of concave functions, expanding understanding of their maximum value structures and applications in interpolation.

Findings

Characterizes the structure of interval maxima as nodes vary.

Provides a homeomorphism theorem for sums of translates.

Applies results to interpolation problems and approximation theory.

Abstract

For a fixed positive integer $n$ consider continuous functions $ K_1,\dots$, $ K_n:[-1,1]\to \mathbb{R}\cup\{-\infty\}$ that are concave and real valued on $[-1,0)$ and on $(0,1]$, and satisfy $K_j(0)=-\infty$. Moreover, let $J:[0,1]\to \mathbb{R}\cup\{-\infty\}$ be upper bounded and such that $[0,1]\setminus J^{-1}(\{-\infty\})$ has at least $n+1$ elements, but it is arbitrary otherwise. For $x_0:=0<x_1<\dots< x_n \le x_{n+1}:=1$, so called nodes, and for $t\in [0,1]$ consider the sum of translates function $F(x_1,\ldots,x_n,t):=J(t)+\sum_{j=1}^n K_j(t-x_j)$, and the vector of interval maximum values $m_j:=m_j(x_1,\ldots,x_n):=\max_{t\in [x_j,x_{j+1}]}F(x_1,\ldots,x_n,t)$ ($j=0,1,\ldots,n$). We describe the structure of the arising interval maxima as the nodes run over the $n$-dimensional simplex. Applications presented here range from abstract moving node Hermite-Fej\'er interpolation for generalized algebraic and trigonometric polynomials via Bojanov's problem to more abstract results of interpolation theoretic flavour.