---

Statement

Setting (printed p. 221). The $n$th row of the node matrix $A$ satisfies (1.1), $1\ge x_{1n}>x_{2n}>\cdots>x_{nn}\ge-1$, and $l_{jn}(x,A)=\omega_n(x,A)/(\omega_n'(x_{jn},A)(x-x_{jn}))$ with $\omega_n(x,A)=\prod_{j=1}^n(x-x_{jn})$ are the ordinary fundamental polynomials of Lagrange interpolation, of degree $n-1$.

Theorem II, first form ((3.13), printed p. 225). For all matrices $A$,

$$\max_{-1\le x\le1}\sum_{j=1}^n|l_{jn}(x,A)|\ge\frac2\pi\log n-c_5\log\log n,$$

where $c_5$ is a positive numerical constant. The paper names (3.13) "Theorem II" on the same page and notes that a somewhat weaker inequality is in S. Bernstein's 1931 paper (its [1]).

Theorem II, restated (printed p. 233, where its proof is sketched). For $n>c_{18}$,

$$\max_{-1\le x\le1}\sum_{\nu=1}^n|l_\nu(x)|>\frac2\pi\log n-c_{19}\log\log n,$$

in the notation, set at the end of § 3 (p. 225), that drops the index $n$ and the matrix $A$. The restated form adds the range $n>c_{18}$, makes the inequality strict and renames the constant $c_{19}$.

Consequence (p. 225). The paper pairs (3.13) with the fact that, for $n>n_1(\varepsilon)$, $\max_{-1\le x\le1}\sum_{j=1}^n|l_{jn}(x,T)|\le(\frac2\pi+\varepsilon)\log n$ for the Chebyshev matrix $T$, and says that together they solve asymptotically the problem of minimizing $\max_{-1\le x\le1}\sum_j|l_{jn}(x,A)|$ over $A$; that is, the minimum is $(\frac2\pi+o(1))\log n$. The paper then declines to formulate the analogues of its questions (3.10)--(3.12) with $l_{jn}$ in place of $\mathfrak{h}_{jn}$; no local ordinary-Lagrange question is printed in this paper.

Source. P. Erdős and P. Turán, An extremal problem in the theory of interpolation, Acta Math. Acad. Sci. Hungar. 12 (1961), 221--234; (3.13) and its consequence on p. 225, the restatement and sketch in § 10, pp. 233--234. The copy read is identified in the source digest.

Read depth. Claims checked: (3.13), the restatement and the consequence were read clause by clause on the page images. The sketch (pp. 233--234) was read on the page images for structure only; no estimate was checked, and nothing here is independently reviewed.

Proof pointer

The paper sketches the proof in § 10 (pp. 233--234) and leaves its last part to the pattern of Theorem I. It first assumes, without loss of generality, $|l_\nu(x)|\le\log n$ on $[-1,1]$ for all $\nu$ (10.1), which gives the equidistribution (6.5) of the node angles. Two cases remain, with $M=\max_{[-1,1]}|\omega|$ and $M_0$ its maximum on the central interval $d_0$ of § 5. If $M_0<M/(20\log^2n)$ (10.2), Lemma I bounds $|\omega'|$ on $d'_0$ and the sum exceeds $\frac54\log n$. If not (10.3), an index $\nu_1\le R$ with $M_{\nu_1+1}\le M_{\nu_1}(1+1/\log n)$ exists (10.4), Lemma I bounds $|\omega'(x_j)|$ on $d'_{\nu_1}$, and the paper says that the rest runs exactly as for Theorem I and drops it. Not checked here.

Dependencies

Lemma I (p. 225) of the same paper, the interval construction of § 5 (pp. 227--228), and the equidistribution (6.5) (p. 229), which for Theorem I rests on P. Erdős, Ann. of Math. 43 (1942), 59--64 (the paper's [3]); the remaining steps are those of Theorem I's Case III (§§ 8--9, pp. 229--232). The Chebyshev upper estimate on p. 225 is stated without a reference.

Bears on

• Problem 1129: Theorem II gives the size $(\frac2\pi+o(1))\log n$ of the minimal Lebesgue constant on $[-1,1]$ (with the Chebyshev upper estimate, p. 225); it does not describe the minimizing nodes, which is what the problem asks.
• Problem 1132: Theorem II bounds the maximum over $x\in[-1,1]$ of $L_n(x)$ for every large $n$, with a $\log\log n$ loss in place of $O(1)$; it says nothing about a fixed point $x$ or about almost every $x$, which the problem's two questions concern.
• Problem 1153: the restated form gives $\max_{[-1,1]}\lambda(x)>(\frac2\pi-o(1))\log n$, the case $a=-1$, $b=1$ of the problem's inequality; the paper sets no subinterval question for $l_{jn}$ (p. 225), and its interval question (3.12) is about $\mathfrak{h}_{jn}$, recorded on the two-layer record.