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

Abstract

Abstract. Schur's [14] Markov-type extremal problem asks to find the maximum (1) sup −1≤ξ≤1 sup Pn∈B n,ξ,2 |P(1) n (ξ)|, where B n,ξ,2 = {Pn ∈ Bn : P (2) n (ξ) = 0} ⊂ Bn = {Pn : |Pn(x)| ≤ 1 for |x| ≤ 1} and Pn is an algebraic polynomial of degree ≤ n. Erdös and Szegö [3] found that for n ≥ 4 this maximum is attained if ξ = ±1 and Pn ∈ B n,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.

An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

“…4.1. The rational side-solution of the sextic Abel-Pell differential equation Regrettably, we have to point to a flaw in the paper by Grasegger and Vo [10] concerning the degree n = 6: The one-parameter power form representation as given there in Section 4.5, and identically given in Section 4.6 (Example 4.1), expressed there as T 3 (Z 2 (x)), which is in fact a rational solution of the sextic Abel-Pell differential equation (29), does not represent, as is claimed in [10], a family of sextic normalized proper Zolotarev polynomials. The reason is that for each parameter t > 1 the sextic polynomial T 3 (Z 2 (x)) equioscillates less than six times (in fact four times) on I. Here,…”

“…We leave it to the reader to reverify the above properties with a method of own choice. It is readily seen that it satisfies, for example, the conditions (17), ( 21), ( 24), ( 27), (28). The graph of Z 6,t=−0.05 , whose uniform norm on I and on [α, β] is 1, is displayed in Figure 1…”

“…Our main result is the representation of the family of normalized proper Zolotarev polynomials of degree 6 in the parameterized power form (5). The parametrization for the cases 2 ≤ n ≤ 4 is a rational one, see [10] and [29], whereas for the case n = 5 it is a radical one, and it also turns out to be so for the case n = 6, see the even-indexed coefficients in Theorem 1 below. We have achieved our result by using symbolic computation (Quantifier Elimination, Cylindrical Algebraic Decomposition, Groebner Basis) as implemented in Mathematica and by using the Algebraic Curve Package algcurves in Maple TM [16].…”

“…Such polynomials of degree 2 ≤ n ≤ 4 and represented in a power form are scattered in the literature, see [5], [7], [10], [22, p. 156], [27], [28], [29], [34] and [39, p. 98] (the latter with respect to [0, 1]). They can be expressed (possibly after some rearrangements) analytically as…”

“…They find application (after rescaling) e.g. in the proof of the Markov inequality [17], [19] and Landau-Kolmogorov inequality [35], in the proof of Schur's Markov-type problem [8], [28], [29] and in the problem of maximizing linear coefficient functionals [27].…”