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This article is cited in 1 scientific paper (total in 1 paper)
Intertwining of maxima of sum of translates functions with nonsingular kernels
B. Farkasa, B. Nagyb, Sz. Gy. Révészc a University of Wuppertal
b Bolyai Institute, University of Szeged
c Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest
Abstract:
In previous papers we investigated so-called sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb{R}}:=\mathbb R\cup\{-\infty\}$ is a “sufficiently nondegenerate” and upper-bounded “field function”, and $K:[-1,1]\to \underline{\mathbb{R}}$ is a fixed “kernel function”, concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0} K(t)=-\infty$. For node systems $\mathbf{x}:=(x_1,\ldots,x_n)$ with $x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j(\mathbf{x}):=\sup_{x_j\le t\le x_{j+1}} F(\mathbf{x},t)$. Among other results we proved a strong intertwining property: if the kernel is decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i(\mathbf{x})$<$m_i(\mathbf{y})$ and $m_j(\mathbf{x})>m_j(\mathbf{y})$. Here we partially succeed to extend this even to nonsingular kernels.
Keywords:
minimax problems; kernel function; sum of translates function; vector of local maxima; equioscillation; intertwining of interval maxima.
Received: 31.07.2022 Revised: 17.10.2022 Accepted: 24.10.2022
Citation:
B. Farkas, B. Nagy, Sz. Gy. Révész, “Intertwining of maxima of sum of translates functions with nonsingular kernels”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 4, 2022, 262–272
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https://www.mathnet.ru/eng/timm1968 https://www.mathnet.ru/eng/timm/v28/i4/p262
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