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Source. Relations (6) and (7) and the question following them, p. 68, 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

Notation as in relations (1)-(2); a point group is a triangular array of nodes $-1\le x_1^{(n)}<\cdots<x_n^{(n)}\le1$, $n=1,2,\ldots$, with fundamental functions $l_k^{(n)}$.

Relation (6) (p. 68). Let $\varepsilon>0$ and $-1\le a<b\le1$. Then if $n>n_0(\varepsilon,a,b)$ and $-1\le x_1<\cdots<x_n\le1$,

$$\max_{a<x<b}\Bigl|\sum_{k=1}^n l_k(x)\Bigr|>\Bigl(\frac2\pi-\varepsilon\Bigr)\log n.$$

As printed the absolute value bars enclose the whole sum. Since $\sum_{k=1}^n l_k(x)=1$ identically, the sum read literally is $1$ and the inequality is meant for the Lebesgue function $\sum_{k=1}^n|l_k(x)|$, as in (7) below; this is a reading of the print, not a correction it records.

Erdős calls the proof of (6) complicated and unpublished, and says that it sharpens a previous result of S. Bernstein.

Relation (7) (p. 68). (6) immediately implies that for any point group $x_i^{(n)}$ there is an $x_0$, $-1<x_0<1$, with

$$\limsup_{n\to\infty}\frac1{\log n}\sum_{k=1}^n\bigl|l_k^{(n)}(x_0)\bigr|\ge\frac2\pi,$$

and in fact the set of such $x_0$ is everywhere dense. Erdős adds: "Perhaps (7) holds for almost all $x_0$."

Question (p. 68). Erdős writes that it would be of interest to know whether for every point group there is an $x_0$ in $(-1,+1)$ for which

$$\sum_{k=1}^n\bigl|l_k^{(n)}(x_0)\bigr|>\frac{2\log n}{\pi}-c$$

holds for infinitely many values of $n$, and that this question does not seem to be easy. The constant $c$ is not further specified.

Read depth. Read clause by clause on the printed page. The paper gives no proof of (6); the step from (6) to (7) is stated as immediate.

Bears on

• Problem 1132: source. The question above and the suggestion that (7) holds for almost all $x_0$ are the problem's two questions, posed here for an arbitrary point group, of which a single sequence of nodes is a special case. Relation (7) gives a dense set of points with the $\limsup$ at least $2/\pi$; it does not give almost all points, nor the additive constant the first question asks for.
• Problem 1153: statement announced without proof. Relation (6), read for the Lebesgue function, is the inequality the problem asks for on a fixed subinterval; the paper states it as Erdős's result but calls the proof complicated and unpublished, and gives none.