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

Rack, Heinz-Joachim; Vajda, Robert

Serdica Journal of Computing (2014)

- Volume: 8, Issue: 1, page 71-96
- ISSN: 1312-6555

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topRack, 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 -