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.

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We consider optimal Lagrange interpolation with polynomials of degree at most two on the unit interval [−1, 1]. In a largely unknown paper, Schurer (1974, Stud. Sci. Math. Hung. 9, 77-79) has analytically described the infinitely many zero-symmetric and zero-asymmetric extremal node systems −1 ≤ x1 < x2 < x3 ≤ 1 which all lead to the minimal Lebesgue constant 1.25 that had already been determined by Bernstein (1931, Izv. Akad. Nauk SSSR 7, 1025-1050). As Schurer’s proof is not given in full detail, we formally verify it by providing two new and sound proofs of his theorem with the aid of symbolic computation using quantifier elimination. Additionally, we provide an alternative, but equivalent, parameterized description of the extremal node systems for quadratic Lagrange interpolation which seems to be novel. It is our purpose to bring the computer-assisted solution of the first nontrivial case of optimal Lagrange interpolation to wider attention and to stimulate research of the higher-degree cases. This is why our style of writing is expository.

Abstract.For evaluation schemes based on the Lagrangian form of a polynomial with degree n, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is between n 2 and n 2 + (3n + 7 )2 n, where 2n denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible order O(n 2) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points.

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