Homeomorphism theorem for sums of translates on the real axis
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914284548653056 |
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| author | Nikiforova, Tatiana M. |
| author_facet | Nikiforova, Tatiana M. |
| contents | 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 $\mathbb{R}^n.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homeomorphism theorem for sums of translates on the real axis Nikiforova, Tatiana M. Classical Analysis and ODEs 41A50, 41A52, 42A15, 26A51 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 $\mathbb{R}^n.$ |
| title | Homeomorphism theorem for sums of translates on the real axis |
| topic | Classical Analysis and ODEs 41A50, 41A52, 42A15, 26A51 |
| url | https://arxiv.org/abs/2509.07776 |