Homeomorphism theorem for sums of translates on the real axis

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 $\mathbb{R}^n.$

💡 Research Summary

The paper studies a class of functions on the real line that are sums of translates of concave kernel functions together with an external field. For a vector of ordered nodes (\mathbf y=(y_1,\dots ,y_n)) with (y_1\le\cdots\le y_n) the authors define
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