📝 SummarXiv

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 $\mathbb{R}^n.$