An example of optimal nodes for interpolation

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

“…If we consider optimal polynomial Lagrange interpolation with n ≥ 4, then very little seems to be known about the analytic expressions of extremal node systems and minimal Lebesgue constants. At least for the cubic case n = 4 the analytic form of the unique extremal and canonical node system and of the minimal Lebesgue constant Λ * 4 has been determined, see [21], [22]. a prospective paper [23] we intend, with the aid of symbolic computation, to shed some more light on the cubic case and we hope to achieve some progress for the next low-degree polynomial cases as well.…”

“…We also take the opportunity to correct a misprint in the footnote of [2, p. 1027]: The node polynomial (x − x 1 )(x − x 2 )(x − x 3 ), denoted there by A n+1 (x), n = 2, should have read (21) A n+1 (x) = x x 2 − 8 9…”

Errata and Addenda to Mathematical Constants

“…If, for n = 4, we restrict x 1 = −1, x 4 = 1 and x 2 = −x 3 , then the smallest Λ 4 corresponds to x * 3 = 0.4177913013... with minimal polynomial 25z 6 + 17z 4 + 2z 2 − 1; it also has value Λ * 4 = 1.4229195732... with minimal polynomial 43w 3 − 93w 2 + 53w − 11. In contrast, Λ * 2 = 1 and Λ * 3 = 5/4 trivially, but Λ * 5 = 1.5594902098... nontrivially with minimal polynomial of degree 73 [510,511,512,513].…”

On node distributions for interpolation and spectral methods

“…where P * (x) is the best polynomial approximation of degree s and Λ(c) is the Lebesgue constant corresponding to the node distribution c = (c i ) s i=0 . The Lebesgue constant Λ(c) indicates how far the Lagrange interpolation polynomial L c (x) is from the best polynomial approximation of degree s. Lebesgue constants have been studied extensively in the literature (see, e.g, [1], [2], [4], [6], [7], [8], [10], and references therein). It is of interest to find a node distribution for which the Lebesgue constant is minimal among all node distributions with the same number of nodes.…”