Minimax problems and results for sum of translates functions

Автори: Bálint Farkas, Béla Nagy and Szilárd Révész, Alfréd Rényi Institute of Mathematics
Дата: 06.06.2023 г.
Час: 17:30 ч.
Място: Зала 503 на ИМИ-БАН

Резюме: We introduce a general framework to investigate minimax problems for sum of translates functions

F(x,t)=\sum_k K(t-x_j) or F(x,t)=J(t)+\sum_k K(t-x_j),

where K is a general concave “kernel function” and J is an “outer field” function, t runs [0,1], and x=(x_1,…,x_n) is a set of nodes which are used to translate the kernel.

Our setup is very close to logarithmic potential theory, but fixing n and focusing on a more detailed analysis, we obtain new results even for very classical problems. Extending a method of P. Fenton, we reach considerable generality while proving precise results for minimax, maximin and equioscillating node systems, and the behavior of the local (interval) maxima. One of our main motivations is to generalize Bojanov’s theorem about the variant of the Chebyshev problem with prescribed zero multiplicities. In particular, we derive a weighted generalization with surprisingly general weights.

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