---

Statement

Setting (printed pp. 221 and 223). $A$ is an infinite triangular matrix whose $n$th row $x_{1n},\ldots,x_{nn}$ satisfies (1.1), $1\ge x_{1n}>x_{2n}>\cdots>x_{nn}\ge-1$. With $\omega_n(x,A)=\prod_{j=1}^n(x-x_{jn})$ (1.2) and the ordinary fundamental polynomials $l_{jn}(x,A)=\omega_n(x,A)/(\omega_n'(x_{jn},A)(x-x_{jn}))$ (1.3), the paper defines in (3.2) the polynomials

$$\mathfrak{h}_{jn}(x,A)=\frac{\omega_n(x,A)^2}{\omega_n'(x_{jn},A)^2(x-x_{jn})} =(x-x_{jn})\,l_{jn}(x,A)^2,$$

the coefficients of the prescribed slopes $y'_{jn}$ in the Hermite interpolation polynomial of degree $\le2n-1$ of (2.5)--(2.6), and in (3.1)

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

The paper writes these polynomials with a Fraktur $\mathfrak{h}$; they are not the ordinary Lagrange polynomials $l_{jn}$, and $M_n(A)$ is not a Lebesgue constant.

Theorem I (printed p. 224). For every choice of the matrix $A$,

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

where $c_1$ is a positive numerical constant (the paper's convention for $c_1,c_2,\ldots$, stated on the same page). The statement prints no range of $n$; the proof closes (p. 232) with the strict bound $\frac2\pi\frac{\log n}n-c_{16}\frac{\log\log n}n$ for $n>c_{17}$.

Consequence (3.7)--(3.8) (p. 224). With $g(n)=\min_AM_n(A)$ (3.5, p. 223) and Fejér's estimate (3.4), $M_n(T)<(\frac2\pi+\varepsilon)\frac{\log n}n$ for $n>n_0(\varepsilon)$, where the $n$th row of $T$ is the roots of the $n$th Chebyshev polynomial, Theorem I gives

$$\lim_{n\to\infty}\frac{n}{\log n}\,g(n)=\frac2\pi .$$

So the Chebyshev matrix $T$ is asymptotically extremal for $M_n$.

Source. P. Erdős and P. Turán, An extremal problem in the theory of interpolation, Acta Math. Acad. Sci. Hungar. 12 (1961), 221--234; the definitions on pp. 221 and 223, Theorem I and (3.7)--(3.8) on p. 224. The copy read is identified in the source digest.

Read depth. Claims checked: the definitions, Theorem I and (3.8) were read clause by clause on the page images. The proof (§§ 4--9, pp. 225--232) was read on the page images for structure only; no estimate was checked, and nothing here is independently reviewed.

Proof pointer

The proof (pp. 225--232) drops the index $n$ and proves the reformulation (3.14) on p. 225. Two lemmas come first (§ 4): Lemma I (p. 225) bounds the derivative of a polynomial of degree $n$ that is at most $M$ on $[-1,1]$ and at most $\eta_1M$ on $[-b,b]$, on a slightly shorter interval, using M. Riesz's interpolation formula; Lemma II (p. 227) uses Markov's inequality to find an interval of length $1/(2m^2)$ on which a polynomial of degree $\le m$ keeps half its maximum. § 5 (pp. 227--228) splits a neighbourhood of $0$ into nested intervals $d_\nu$ of half-length about $(1/\log n)(1+1/\log^2n)^\nu$ and records the maxima $M_\nu$ of $|\omega|$ on them. The proof then has three cases. Case I (§ 6, p. 228): some $|l_{k_0}|$ reaches $n^3$ on $[-1,1]$, and Lemma II gives the bound directly; otherwise every $|l_k|<n^3$, which by Erdős's 1942 theorem on the uniform distribution of roots (the paper's [3]) gives the equidistribution (6.5) of the angles $\vartheta_j$. Case II (§ 7, p. 229): $M_0<M/\log^2n$, where Lemma I bounds $|\omega'|$ on $d'_0$. Case III (§§ 8--9, pp. 229--232): an index $\nu_0$ with slow growth $M_{\nu_0+1}\le M_{\nu_0}(1+1/\log n)$ exists, Lemma I bounds $|\omega'(x_j)|$ on two consecutive intervals, and summing $1/|\Theta_{\nu_0}-\vartheta_j|$ with (6.5) yields the harmonic sum that gives $\frac2\pi\log n$. Not checked here.

Dependencies

Lemma I uses M. Riesz's trigonometric interpolation formula (the paper's [9]); Lemma II uses A. Markov's inequality ([8]); the equidistribution (6.5) rests on P. Erdős, On the uniform distribution of the roots of certain polynomials, Ann. of Math. 43 (1942), 59--64 ([3]), which footnote 11 (p. 228) calls an improvement of the proof in Erdős and Turán, Ann. of Math. 41 (1940), 510--553 (the paper's [4]), especially pp. 548--552. The upper estimate (3.4) is Fejér's (Math. Z. 32 (1930), the paper's [7]).

Bears on

Theorem I concerns the derivative-data polynomials $\mathfrak{h}_{jn}$ and the scale $\log n/n$, not the Lebesgue function $\sum_j|l_{jn}(x)|$ of the catalog's interpolation problems, so it bears on none of them directly. The ordinary-Lagrange analogue is Theorem II, whose sketched proof follows this one; the two-layer record keeps the questions (3.10)--(3.12) that the paper attaches to Theorem I.