Minimax and maximin problems for sums of translates on the real axis

TL;DR

This paper solves minimax and maximin problems for sums of translates on the real axis, establishing the uniqueness of extremal functions and reducing the problem to a segment, with an analogue of the Mhaskar-Rakhmanov-Saff theorem.

Contribution

It provides the first solution to these problems for sums of translates and proves the uniqueness of the extremal function, extending classical approximation theory.

Findings

Unique extremal function exists for both problems

Reduction to a segment simplifies the problem

Analogue of Mhaskar-Rakhmanov-Saff theorem developed

Abstract

Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar-Rakhmanov-Saff theorem, too.