A minimax problem for sums of translates on the torus. (4th January 2018) - Farkas, Bálint; Nagy, Béla; Révész, Szilárd Gy.
Abstract: We extend some equilibrium‐type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T ≃ [ 0, 2 π ), but the motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton. The problem is to minimize — with respect to the arbitrary translates y 0 = 0, y j ∈ T, j = 1, ⋯, n — the maximum of the sum function F : = K 0 + ∑ j = 1 n K j ( · − y j ), where the functions K j are certain fixed 'kernel functions'. In our setting, the function F has singularities at functions y j, while in between these nodes it still behaves regularly. So one can consider the maxima m i on each subinterval between the nodes y j, and minimize max F = max i m i . Also the dual question of maximization of min i m i arises. Hardin, Kendall and Saff considered one even kernel, K j = K for j = 0, ⋯, n, and Fenton considered the case of the interval [ − 1, 1 ] with two fixed kernels K 0 = J and K j = K for j = 1, ⋯, n . Here we build up a systematic treatment when all the kernel functions can be different without assuming them to be even. As an application we generalize a result of Bojanov about Chebyshev‐type polynomials with prescribed zero order.