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Claim. Theorem 1 of P. Erdős, Problems and results on the theory of interpolation. II, Acta Math. Acad. Sci. Hungar. 12 (1961), 235--244 (received 7 July 1960; card Erdős 1961 II), states, on p. 235, that for distinct nodes $-1\le x_1<\cdots<x_n\le1$

$$\max_{-1\le x\le1}\sum_{k=1}^n\lvert l_k(x)\rvert>\frac2\pi\log n-c_1,$$

with $c_1$ an absolute positive constant. With $\varepsilon_n=(c_1+1)/\log n\to0$ this gives $\max_{[-1,1]}\lambda>(2/\pi-\varepsilon_n)\log n$ for every node configuration, which answers the case $a=-1$, $b=1$ of Problem 1153 yes. The theorem improves Theorem II of Erdős and Turán, An extremal problem in the theory of interpolation, in the same volume, pp. 221--234, equation (3.13), p. 225 (card Erdős and Turán 1961), which gives the bound $(2/\pi)\log n-c_5\log\log n$ and so already settles the same instance; that paper only sketches its proof of Theorem II, on pp. 233--234, and it is named on this page rather than paged, since the later theorem supersedes it. On the same p. 235 Erdős writes that Bernstein asserted the sharp whole-interval bound and proved it in full only for trigonometric interpolation, and that he could not reconstruct Bernstein's proof of the algebraic case.

Covers. The case $a=-1$, $b=1$ of the question.

Acceptance. Refereed: Acta Mathematica Academiae Scientiarum Hungaricae, volume 12 (1961), pp. 235--244. The site's PROVED label credits Tao, so its commentary's mention of this bound gives no reviewed evidence. No check of the proof is recorded.

Depends on. Nothing in this wiki.