The first Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur

Abstract

Schur’s [14] Markov-type extremal problem asks to find the maximum (1) (2) (1)  sup  sup |Pn  (ξ)|, where Bn,ξ,2 = {Pn ∈ Bn : Pn  (ξ) = 0} ⊂ Bn =−1≤ξ≤1 Pn ∈Bn,ξ,2 {Pn : |Pn(x)| ≤ 1 for |x| ≤ 1} and Pn is an algebraic polynomial of degree ≤ n. Erdo¨s and Szego¨ [3] found that for n ≥ 4 this maximum is attained if ξ = ±1 and Pn ∈ Bn,ξ,2 is a (unspecified) member of the 1-parameter family of hard-core Zolotarev polynomials Zn,t. Our first result centers around the proof in [3] for the initial case n = 4 and is three-fold: (i) the numerical value for (1) in ([3], (7.9)) is not correct, but sufficiently precise; (ii) from preliminary work in [3] can in fact be deduced a closed analytic expression for (1) if n = 4, allowing numerical evaluation to any precision; (iii) even the explicit power form representation of an extremal Z4,t = Z4,t∗ can be deduced from [3], thus providing an exemplification of Schur’s problem that seems to be novel. Additionally, we cross-check its validity twice: firstly by deriving Z4,t∗ conversely from a general formula for Z4,t that we have given in [12], and secondly by applying Theorem 5.10 in [11]. We then turn to a generalized solution of Schur’s problem, to k -th derivatives, by Shadrin [16]. Again we provide in explicit form the corresponding maximum as well as an extremizer polynomial for the first non-trivial degree n = 4. Finally, we contribute to the fuller description of Z4,t by providing its critical points in explicit form.

Mathematics Subject Classification (2010): 26C05, 26D10, 41A10, 41A17, 41A29, 41A44, 41A50.