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Source. P. 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. The problems are unnumbered; the second one is displayed as (3).

Statement

Notation as in relations (1)-(2): nodes $-1\le x_1<\cdots<x_n\le1$, fundamental functions $l_k$, and Lebesgue function $\sum_{k=1}^n|l_k(x)|$; put $x_0=-1$ and $x_{n+1}=1$.

First problem (p. 66). Erdős poses as an interesting unsolved problem the task to determine the set $-1\le x_1<\cdots<x_n\le1$ for which

$$\max_{-1\le x\le1}\sum_{k=1}^n|l_k(x)|$$

is minimal. He writes that it "seems likely" that this set is characterized by the property that the values of the $n+1$ local maxima of $\sum_{k=1}^n|l_k(x)|$ are all equal (with $-1=x_0$, $1=x_{n+1}$), and that as far as he knows this conjecture is still unproved. He suggests the conjecture may be easier on the unit circle: for nodes $x_i$ on the unit circle, minimizing $\max_{|z|=1}\sum_{k=1}^n|l_k(z)|$, he writes that it seems certain that the $x_i$ must be the $n$-th roots of unity.

Problem (3) (p. 66). Determine the set $-1\le x_1<\cdots<x_n\le1$ for which

$$\min_{0\le i\le n}\ \max_{x_i<x<x_{i+1}}\ \sum_{k=1}^n|l_k(x)| \qquad(3)$$

is maximal. Erdős writes that it seems likely that the two problems have the same solution, again with the $n+1$ maxima equal. He cannot prove the analogue of (2) for (3); he can only show (his reference [5], P. Erdős, Some remarks on polynomials, Bull. Amer. Math. Soc. 53 (1947), 1169-1176) that

$$\min_{0\le i\le n}\ \max_{x_i<x<x_{i+1}}\ \sum_{k=1}^n|l_k(x)|<\sqrt n,$$

and writes that it seems certain that $\sqrt n$ can be replaced by $c_3\log n$.

Read depth. Read clause by clause on the printed page. The paper proves nothing here; the $\sqrt n$ bound is cited from [5].

Bears on

• Problem 1129: source. The first problem is the question the problem page states, and the equal-maxima property is Erdős's conjectured description of its answer; the paper offers it as likely and proves nothing toward it.
• Problem 1130: source. Problem (3) is the quantity the problem studies, with $x_0=-1$, $x_{n+1}=1$; the paper asks for its maximizing nodes, proves the bound below $\sqrt n$ (cited from [5]), and expects $c_3\log n$, which is the problem's first question. It proves no logarithmic bound.