Erdos 1961 extremal problem theory interpolation

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theorem_i: For every node matrix the largest sum of the absolute values of the derivative-data Hermite polynomials is at least (2/(πn))(log n − c_1 log log n), so Chebyshev nodes are asymptotically optimal for this quantity.

theorem_ii: For every node matrix the Lebesgue constant of ordinary Lagrange interpolation on [−1,1] is at least (2/π) log n − c_5 log log n, so Chebyshev nodes are asymptotically optimal; the proof is sketched.

two_interpolation_layers: Separates the two source quantities and records the ordinary-Lagrange bridge on p. 225.

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P. Erdős and P. Turán, An extremal problem in the theory of interpolation, Acta Math. Acad. Sci. Hungar. 12 (1961), 221--234. Primary source. The copy read for this card is the 14-page scan at that address. It carries no notice; the publisher's page for the journal's backfile, read for a 1978 article in the same journal, shows "© Akadémiai Kiadó" under "Reprints and permissions" with subscription access (https://link.springer.com/article/10.1007/BF01902213), and this article's own page (DOI 10.1007/BF02066685) was not consulted; every other right reserved.

This is the Erdős--Turán article. The distinct Erdős solo paper Problems and results on the theory of interpolation. II, pp. 235--244, is filed separately at its source home.

The source’s first layer concerns the derivative-data Hermite polynomials $\mathfrak{h}_{jn}(x,A)=(x-x_{jn})l_{jn}(x,A)^2$. Their maximum absolute sum has scale $\log n/n$. Theorem I and (3.8), p. 224, establish the asymptotic extremality of Chebyshev nodes for this quantity; (3.10)--(3.12) are unproved questions in the paper. The source writes these polynomials with a Fraktur $\mathfrak{h}_{jn}$, defined in (3.2), p. 223.

The second layer, on p. 225, is ordinary Lagrange interpolation: equation (3.13), called Theorem II, gives the global lower bound $(2/\pi)\log n-c_5\log\log n$ for $\max_x\sum_j|l_{jn}(x,A)|$, for all matrices $A$. The authors then omit formulations analogous to (3.10)--(3.12) with $l_{jn}$ in place of $\mathfrak{h}_{jn}$. This ordinary layer explains the relationship to the local question E1153 and the global/pointwise interpolation family; the displayed (3.12) must not itself be identified with E1153’s ordinary Lebesgue function.

The two-layer statement record preserves the exact definitions, normalizations, locators and historical question scope. Complete proof compilation of Theorems I and II remains pending; their source text is public at the address above. No historical conjecture status or modern sharp local theorem is inferred from the derivative-data result.

Bears on. Problem 1129: Theorem II ((3.13), p. 225), with the Chebyshev upper estimate printed beside it, gives the size $(\frac2\pi+o(1))\log n$ of the minimal Lebesgue constant on $[-1,1]$; it does not describe the minimizing nodes the problem asks for. Problem 1132: Theorem II bounds the maximum over $x\in[-1,1]$ for each large $n$, with a $\log\log n$ loss in place of $O(1)$; it says nothing about a fixed $x$ or almost every $x$. Problem 1153: Theorem II as restated on p. 233, $>\frac2\pi\log n-c_{19}\log\log n$ for $n>c_{18}$, implies the case $a=-1$, $b=1$ of the problem's inequality; the paper omits the ordinary local formulations (p. 225), and its interval question (3.12) concerns $\mathfrak{h}_{jn}$, not the Lebesgue function. Theorem I bears on none of the three directly.

Results.

• Theorem I (p. 224): for every $A$, $\max_{-1\le x\le1}\sum_j|\mathfrak{h}_{jn}(x,A)|\ge\frac{2}{\pi n}(\log n-c_1\log\log n)$, hence, with Fejér's estimate (3.4), $\lim_{n\to\infty}(n/\log n)g(n)=2/\pi$ (3.8).
• Theorem II ((3.13), p. 225; restated p. 233): for every $A$, $\max_{-1\le x\le1}\sum_j|l_{jn}(x,A)|\ge\frac2\pi\log n-c_5\log\log n$; the proof is sketched on pp. 233--234.
• The two layers: the definitions, the questions (3.10)--(3.12) and the bridge between the derivative-data and ordinary settings.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.