---

Source. H.-J. Rack and R. Vajda, Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant, Stud. Univ. Babeş-Bolyai Math. 60 (2015), no. 2, 151--171; the edition read is named on the source card.

Statement

Setting (pp. 154-155). For a node system $X_n$ in $\mathbf I=[-1,1]$, the Lebesgue function $\lambda_n$ has exactly one local maximum $\mu_i$ in each interval $(x_i,x_{i+1})$, $1\le i\le n-1$ (Proposition 2.2 ii, p. 154). A canonical node system (CNS) is one with $x_1=-1$ and $x_n=1$ (Definition 2.7, p. 155).

Proposition 2.8 (p. 155). If the Lebesgue function of a CNS $X_n$ satisfies the equioscillation property

$$\mu_1=\mu_2=\cdots=\mu_{n-2}=\mu_{n-1},\qquad(2.11)$$

then $X_n$ is an extremal node system, so $\Lambda_n(X_n)=\Lambda_n^*$.

The paper introduces it as answering a conjecture going back to Bernstein (its reference [3]) and as proved in its references [8] and [14], de Boor and Pinkus (J. Approx. Theory 24 (1978), 289-303) and Kilgore (J. Approx. Theory 24 (1978), 273-288). It adds (p. 155) that those papers also prove that a CNS satisfying (2.11) is unique and zero-symmetric, $x_i^*=-x_{n-i+1}^*$.

Proof pointer

Not proved in this paper; it is cited from the references named above.

Read depth

Claims checked: the statement was read on the page image of the print. The cited proofs were not read for this page. Nothing here is independently reviewed.

Dependencies

External: de Boor and Pinkus 1978 and Kilgore 1978, as cited by the paper. The paper uses the uniqueness of the optimal CNS in the proof of Theorem 5.2.

Bears on

• Problem 1129: the paper records, as known from its references [8] and [14], that among node systems containing both endpoints, one whose Lebesgue function equioscillates attains the minimal Lebesgue constant. The paper does not prove it.