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Statement

Theorem XII (p. 545). Suppose that for the matrix $\mathfrak M$

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

for $n>n_{10}(\epsilon)$ (the hypothesis being made for every $\epsilon>0$, as in (69)). Then for every $0\le\alpha<\beta\le\pi$

$$\lim_{n\to\infty}\frac1n\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1 =\frac{\beta-\alpha}{\pi}.$$

Lemma XI (p. 545), which the paper reproduces from M. Riesz: if a trigonometric polynomial $f(\vartheta)$ of order $n$ attains its absolute maximum on $[0,2\pi]$ at $\vartheta_0$, it has no root in $[\vartheta_0-\pi/2n,\vartheta_0+\pi/2n]$. Its corollary: if such a polynomial attains its absolute maximum between two real roots, those roots are at distance at least $\pi/n$.

Proof pointer

Pp. 546--547. If some $[\alpha,\beta]$ held too few nodes along a sequence of $n$, the paper multiplies the node factors in $[\alpha,\beta]$ by extra factors at equally spaced angles outside it, getting a polynomial $G$ of degree less than $(1-c_{68}/3\pi)n$ (72); Lemma XI places its absolute maximum at an angle $\gamma$ inside the interval. A power of $1-(x-\cos\gamma)^2/4$ damps $G$ away from $\cos\gamma$ while keeping the degree below $n$ (73)--(74); interpolating the product at the nodes and using the hypothesis gives an inequality that fails for large $n$.

Read depth

Claims checked: Theorem XII and Lemma XI with its corollary 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 XI (M. Riesz, Jahresbericht der Deutschen Mathematiker-Vereinigung, 1915), as reproduced in the paper.

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.