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Source. Relations (1) and (2), pp. 65-66, of P. Erdős, "Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems," Mathematica (Cluj) 10 (33) (1968), 65-73; see the source card.

Statement

Setting (p. 65). Let $-1\le x_1<\cdots<x_n\le1$ be $n$ points, and let

$$l_k(x)=\frac{\omega(x)}{\omega'(x_k)(x-x_k)},\qquad \omega(x)=\prod_{k=1}^n(x-x_k),$$

be the fundamental functions of Lagrange interpolation. The sum $\sum_{k=1}^n|l_k(x)|$ is the Lebesgue function of the nodes.

Relations (1)-(2) (pp. 65-66). Erdős recalls that he proved (his references [3], [4], sharpening earlier results of Faber, Bernstein and others):

• for every $\varepsilon>0$ there is an $\eta>0$ such that, for $n>n_0(\varepsilon,\eta)$, the set of $x$ with $\sum_{k=1}^n|l_k(x)|<\eta\log n$ has measure less than $\varepsilon$ (relation (1));
• with a constant $c_1$,

$$\max_{-1\le x\le1}\sum_{k=1}^n|l_k(x)|>\frac2\pi\log n-c_1 \qquad(2).$$

He calls both "in some sense best possible" (p. 66), and recalls as well known that if the $x_k$ are the roots of the Chebyshev polynomial $T_n(x)$, then $\max_{-1\le x\le1}\sum_{k=1}^n|l_k(x)|<\frac2\pi\log n+c_2$.

Read depth. The statements were read clause by clause on the printed page. The paper gives no proof; it cites its references [3], [4] (P. Erdős, Problems and results on the theory of interpolation I and II, Acta Math. Acad. Sci. Hungar. 9 (1958), 381-388, and 12 (1961), 235-244).

Bears on

• Problem 1129: background. Relation (2) is a lower bound for the quantity whose minimizing nodes the problem asks to describe; it does not describe the minimizers.
• Problem 1132: background. Relation (2) bounds the maximum over the whole interval, while the problem asks for a single point at which the bound recurs for infinitely many $n$.