On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

Rack, Heinz-Joachim, and Vajda, Robert. "On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation." Serdica Journal of Computing 8.1 (2014): 71-96. <http://eudml.org/doc/268669>.

@article{Rack2014,
abstract = {ACM Computing Classification System (1998): G.1.1, G.1.2.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.},
author = {Rack, Heinz-Joachim, Vajda, Robert},
journal = {Serdica Journal of Computing},
keywords = {Extremal; Interpolation Nodes; Lagrange Interpolation; Lebesgue Constant; Minimal; Optimal; Polynomial; Quadratic; Quantifier Elimination; Symbolic Computation},
language = {eng},
number = {1},
pages = {71-96},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation},
url = {http://eudml.org/doc/268669},
volume = {8},
year = {2014},
}

TY - JOUR
AU - Rack, Heinz-Joachim
AU - Vajda, Robert
TI - On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation
JO - Serdica Journal of Computing
PY - 2014
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 8
IS - 1
SP - 71
EP - 96
AB - ACM Computing Classification System (1998): G.1.1, G.1.2.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.
LA - eng
KW - Extremal; Interpolation Nodes; Lagrange Interpolation; Lebesgue Constant; Minimal; Optimal; Polynomial; Quadratic; Quantifier Elimination; Symbolic Computation
UR - http://eudml.org/doc/268669
ER -