On the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton

TL;DR

This paper extends classical minimax and maximin problems for functions on [0,1], involving sums of translates of kernel functions, revealing new extremal properties and an intertwining phenomenon of interval maxima.

Contribution

It generalizes existing results in extremal function theory by considering a broad class of functions and uncovers novel phenomena in Chebyshev extremal problems.

Findings

Extended minimax and equioscillation results for sums of translates.

Discovered an intertwining phenomenon of interval maxima.

Provided new insights into classical Chebyshev extremal problems.

Abstract

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erd\'elyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.