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

Abstract

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.

A note on optimal Hermite interpolation in Sobolev spaces

“…The most important optimal Lagrange interpolation nodes problem is for C 0 in L ∞ . For n = 3 and n = 4, the results can be found in [11] and [12], respectively. For n ≥ 5, it is still an open problem.…”

Calibration approach to quantify nonlinearity of MEMS pore pressure sensors using optimal interpolation

“…As a result, the number of calibration points should be balanced because it influences calibration time and cost. A small number of points cannot provide appropriate solution, whereas a large number of points increases the time and expense of calibration and data processing [47]. The large number of points also cause a computational overload without generating better linearization [24].…”

“…For example, a higher operating range require more calibration points. There is not a standard method to give an optimum number of points for calibration; however, it is recommended to have two points at zero and full span, and at least two or three subset points for reaching a proper result when full span is high [47]. Thus, a set of five and six points can give appropriate outcome for calibration.…”

“…The accuracy and convergency of polynomial function completely depends on how set of data are chosen [44,45]. A proper selection of calibration points can enhance the accuracy of calibration model, calibration time and cost [46,47]. As simple way, the uniform approach is used for nonlinearity calibration [48,49].…”

“…Distribution of measurement points should be optimised to achieve better outcome and robust calibration [51]. The subject of optimum design for polynomial curve fitting are still an intriguing topic for mathematician since there is no elegant general method for description of optimal matrix [54][55][56]. Betta et al in 2001 provides experimental design for optimizing calibration approach with minimizing calibration points [23].…”