Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 8, No 1, (2014), 71p-96p
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.
ACM Computing Classification System (1998): G.1.1, G.1.2.