---

Claim. Among the node systems $-1=t_0<t_1<\cdots<t_n=1$ containing both endpoints, the canonical convention, to which the free-node question of Problem 1129 reduces by an affine change of variable, the Lagrange interpolating projection of minimal norm, that is the system minimizing the Lebesgue constant $\max_{[-1,1]}\sum_k|l_k(x)|$, is characterized by the equioscillation of its Lebesgue function: the maxima $\lambda_i$ of $\sum_k|l_k(x)|$ on the $n$ gaps between consecutive nodes are all equal. This is Bernstein's conjecture [Be31]; with that reduction it describes the free-node minimizers the problem asks for, so the claim value is answered. The paper is T. A. Kilgore, A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm, J. Approx. Theory 24 (1978), no. 4, 273–288, which continues Kilgore's earlier announcement [Ki77] that a minimizer must equioscillate. This account follows the paper's title, its bibliographic record and the note added in proof of de Boor and Pinkus, which records that Kilgore too had proved Bernstein's conjecture, by an argument along different lines from theirs. The uniqueness of the equioscillating system and the strict comparison with every other system are recorded on the page of de Boor and Pinkus 1978, which appeared in the same issue.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: Journal of Approximation Theory 24 (1978), no. 4, 273–288, in the issue dated December 1978 in the publisher's record, which dates this page. Not reviewed in the sense of this corpus: the site's commentary credits the characterization to Kilgore and Cheney, Kilgore's announcement of 1977 and de Boor and Pinkus, and does not cite this paper, so no curator credit is listed. No formalization declares itself a formalization of this paper.