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Source. Relation (4), p. 67, 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).

Relation (4) (p. 67). For nodes $-1\le x_1<\cdots<x_n\le1$,

$$\int_{-1}^{+1}\sum_{k=1}^n|l_k(x)|\,dx>c_4\log n.$$

Turán's question (p. 67). Turán asked for which set $-1\le x_1<\cdots<x_n\le1$ the integral in (4) is minimal. Erdős writes that the question does not seem easy, but that it seems certain that asymptotically the minimum is attained at the roots of the Chebyshev polynomial $T_n(x)$.

Proof pointer

The paper says only that (1) easily implies (4). The step is that, by (1), the Lebesgue function is at least $\eta\log n$ outside a set of small measure; this gloss is the page's, not the paper's.

Read depth. Read clause by clause on the printed page.

Bears on

No Erdős problem in the corpus. Turán's question is reported here as Turán's, with Erdős's expectation of the answer.