The derivative-data and ordinary Lagrange layers

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Source. P. Erdős and P. Turán, An extremal problem in the theory of interpolation, Acta Math. Acad. Sci. Hungar. 12 (1961), 221--234. The paper uses two distinct interpolation quantities. The following are source-statement records, not complete proof reconstructions.

Hermite derivative-data layer

The nodes in each row of $A$ are distinct, ordered as $1\ge x_{1n}>\cdots>x_{nn}\ge-1$ on p. 221. The ordinary fundamental polynomials are $l_{jn}$. On p. 223, equations (3.1)--(3.2), the source defines

$$\mathfrak{h}_{jn}(x,A)=(x-x_{jn})l_{jn}(x,A)^2, \qquad M_n(A)=\max_{-1\le x\le1}\sum_{j=1}^n|\mathfrak{h}_{jn}(x,A)|.$$

These polynomials multiply derivative data in Hermite interpolation. The symbol in the source is the Fraktur $\mathfrak{h}_{jn}$. The rational expression in (3.2) is understood through this polynomial continuation at a node. It is not the ordinary Lagrange Lebesgue function.

Theorem I on p. 224 states, for every node matrix $A$,

$$M_n(A)\ge\frac{2}{\pi n}(\log n-c_1\log\log n).$$

Together with the Chebyshev upper estimate (3.4), it gives (3.8): $\lim_{n\to\infty}(n/\log n)g(n)=2/\pi$, where $g(n)=\min_A M_n(A)$. The same page presents unproved questions (3.10)--(3.12): an integral lower bound of order $\log n/n$, an almost-everywhere-type lower bound outside a set whose measure tends to zero, and, for fixed $-1\le a<b\le1$,

$$\max_{a\le x\le b}\sum_j|\mathfrak{h}_{jn}(x,A)| >\left(\frac2\pi-\varepsilon\right)\frac{\log n}{n} \quad(n>n_0(\varepsilon,a,b)). \tag{3.12}$$

The authors also ask whether the threshold can depend only on $\varepsilon$, ask for exact small-$n$ values of $g(n)$, and discuss replacing the $\log\log n$ term and proving convexity. These are historical questions in this derivative-data setting, with no status update inferred here.

Ordinary Lagrange layer and the local question

Printed p. 225 / physical p. 5 explicitly turns to the ordinary polynomials:

$$\max_{-1\le x\le1}\sum_{j=1}^n|l_{jn}(x,A)| \ge\frac2\pi\log n-c_5\log\log n. \tag{3.13}$$

It states this for all matrices $A$, calls the result Theorem II, and says that its proof will be sketched. The following prose explicitly omits formulations analogous to (3.10), (3.11) and (3.12) with $l_{jn}$ replacing $\mathfrak{h}_{jn}$. Thus this page supplies the ordinary-Lagrange historical bridge to Problem 1153; (3.12) itself is the displayed derivative-data question with the different $1/n$ normalization. No exact ordinary local formula is printed in that omitted-formulations sentence.

The explicitly printed ordinary local question is also preserved as Va99 item 2.44. Theorem II’s global result and the derivative-data Theorem I are distinct. Their complete source proofs/sketches and same-paper dependencies remain ordinary proof-compilation work; the paper proves Theorem I on pp. 225--232 and sketches the proof of Theorem II on pp. 233--234, where it restates Theorem II for $n>c_{18}$ with a strict inequality and the constant $c_{19}$ in place of $c_5$. Neither historical result replaces Tao’s modern local theorem.