Homeomorphism theorem for sums of translates 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 functions

$$(F\text{y},t) := J(t) + \sum _{j=1}^n K_j(t-y_j), \quad \text{y} := (y1,\ldots,y_n), \ y_1 \le \cdots \le y_n, \ t \in \mathbb{R},$$

where the kernels\(K_1,\ldots,K_n\) are concave on \((-\infty,0)\) and on \((0,\infty)\), having a singularity at \(0\), and \(J\colon \mathbb{R}\to \mathbb{R}\cup\{-\infty\}\) is the field function. We consider "local maxima"

$$m_0(\text{y}) := \sup _{t \in (-\infty, y_1]} F(\text{y}, t), \quad m_n(\text{y}) := \sup _{t \in [y_n, \infty)} F(\text{y}, t),$$

$$m_j(\text{y}) := \sup _{t \in [y_j, y_{j+1}]} F(\text{y}, t), \quad j = 1,\ldots,n-1, $$

and the difference function

$$(D\text{y}) := (m_1\text({y})-m_0\text({y}), m_2\text({y})-m_1\text({y}), \cdots, m_n\text({y})-m_{n-1}\text({y})). $$

We prove that, under certain assumptions on the kernels and the field, \(D\) is a homeomorphism between its domain and \(\mathbb{R}^n\).