Homeomorphism theorem for sums of translates on the real axis

TL;DR

The paper analyzes sums of translates on the real axis, $F(oldsymbol{y},t)=J(t)+ obreak obreak obreak extstyleigl( obreak extstyle obreak K_j(t-y_j)igr)$, and proves that the associated difference map $D$ is a global homeomorphism from the regularity set $R$ to $ obreak obreak $ obreak $ $, under singular, strictly concave kernels with generalized monotonicity, enabling unique minimax solutions on the real axis. The authors reduce the axis problem to the segment, establish local bi-Lipschitzness and properness of $D$, and apply Ho’s global homeomorphism theorem to obtain a global homeomorphism and stability. They further extend these results to the semiaxis, to weighted generalized polynomials, and to interpolation problems for products of log-concave functions, including abstract log-concave interpolation and moving-node Hermite–Fejér interpolation. This yields uniqueness of minimax points and provides a unified framework for minimax interpolation and weighted polynomial models on the real axis.

Abstract

In this paper, we study sums of translates on the real axis. These functions generalize logarithms of weighted algebraic polynomials. Namely, we are dealing with the following functions \[ F(\mathbf{y},t) := J(t) + \sum \limits_{j=1}^n K_j(t-y_j), \quad \mathbf{y} := (y_1,\ldots,y_n), \ y_1 \le \ldots \le y_n, \] where the field function $J$ is a function defined on $\mathbb{R}$, which is "admissible" for the kernels $K_1,\ldots,K_n$ concave on $(-\infty,0)$ and on $(0,\infty)$ and having a singularity at $0.$ We consider "local maxima" \begin{gather*} \begin{aligned} m_0(\mathbf{y}) & := \sup \limits_{t \in (-\infty, y_1]} F(\mathbf{y}, t), \quad m_n(\mathbf{y}) := \sup \limits_{t \in [y_n, \infty)} F(\mathbf{y}, t),\\ m_j(\mathbf{y}) & := \sup \limits_{t \in [y_j, y_{j+1}]} F(\mathbf{y}, t), \quad j = 1,\ldots,n-1, \end{aligned} \end{gather*} and the difference function \[ D(\mathbf{y}) := (m_1(\mathbf{y})-m_0(\mathbf{y}), m_2(\mathbf{y})-m_1(\mathbf{y}),\ldots,m_n(\mathbf{y})-m_{n-1}(\mathbf{y})). \] We prove that, under certain assumptions on monotonicity of the kernels, $D$ is a homeomorphism between its domain and $\mathbb{R}^n.$

Theorems & Definitions (38)

• Definition 1.1
• Definition 1.2
• Theorem A
• Theorem B
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Definition 1.3
• Theorem 1.1
• Lemma 2.1