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Statement

Setting (pp. 510--511): an arbitrary node matrix $\mathfrak M$ with rows $1\ge x_1^{(n)}>\cdots>x_n^{(n)}\ge-1$, node polynomial $\omega_n$ and fundamental functions $l_k$.

Theorem V (p. 532). Suppose that, for every sufficiently small $\epsilon>0$,

$$[|l_k(x)|]^{1/n}\le1+\epsilon,\qquad-1\le x\le1,\quad k=1,\ldots,n,\quad n>n_2(\epsilon) \tag{41}$$

holds. Then at every fixed point $z$ of the complex plane cut along $[-1,1]$

$$\lim_{n\to\infty}[\omega_n(z)]^{1/n}=\frac{z+\sqrt{z^2-1}}{2},$$

the roots being taken positive on the positive real axis for $z>1$.

Lemma VI (p. 532). Under (41), for every small $\eta>0$ and $n>n_3(\eta)$, $|\omega_n'(x_\nu)|>(\frac12-\eta)^n$ for $\nu=1,\ldots,n$. The paper remarks (p. 533), without using it, that Lemma VI and Lemma I give $\lim_n[\sum_\nu1/|\omega_n'(x_\nu)|]^{1/n}=2$ under (41).

The introduction (p. 517) presents the theorem as (20)--(21) and notes that it can also be derived indirectly from a theorem of Kalmár through a remark of Pólya; the paper's proof is direct.

Proof pointer

Pp. 532--534. Lemma VI follows from Chebyshev's theorem that a monic polynomial of degree $n-1$ reaches $2^{-(n-2)}$ in absolute value on $[-1,1]$, applied to $\omega_n(x)/(x-x_\nu)$. For the upper bound, $\omega_n$ is interpolated at the $n+1$ roots of the Chebyshev polynomial $T_{n+1}$, and (41) with Lemma I bounds $|\omega_n|$ on $[-1,1]$. For the lower bound, $T_{n-1}$ is interpolated at the roots of $\omega_n$, which gives (43) and, with Lemma VI, (44)--(45); both sides being one-valued and regular on the cut plane, the limit follows.

Read depth

Claims checked: Theorem V, (41), Lemma VI and the remark were read clause by clause on the page images of the print; the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

Lemma I (stated on the Theorem I page) and Lemma VI of the same paper; Chebyshev's extremal property of $T_n$.

Source. P. Erdős and P. Turán, On interpolation. III. Interpolatory theory of polynomials, Annals of Mathematics (2) 41 (3) (1940), 510--553, DOI 10.2307/1968733; the edition read is named on the source card.

Bears on

None of the problem pages directly.