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Statement

Setting (p. 510): the nodes of the $n$th row are written $x_\nu^{(n)}=\cos\vartheta_\nu^{(n)}$ with $0\le\vartheta_1^{(n)}<\cdots<\vartheta_n^{(n)}\le\pi$ (display (4)).

Theorem VII (p. 536). Let the matrix $\mathfrak M$ be such that $[-1,1]$ contains a subinterval $[b,a]\equiv[\cos\beta,\cos\alpha]$ on which

$$|l_k(x)|\le c_{49}\quad(k=\nu,\nu+1,\ldots,\mu), \qquad |l_k(x)|<c_{50}n^{c_{51}}\quad\text{for the other }k,$$

where $\vartheta_{\nu-1}^{(n)}<\alpha\le\vartheta_\nu^{(n)}<\vartheta_{\nu+1}^{(n)}<\cdots<\vartheta_\mu^{(n)}\le\beta<\vartheta_{\mu+1}^{(n)}$, that is, $\nu,\ldots,\mu$ index the nodes in the subinterval. Then

$$\frac{[\epsilon(b-a)]^{1/2}}{c_{49}}\cdot\frac1n \le\vartheta_{k+1}^{(n)}-\vartheta_k^{(n)} \le\frac{c_{49}\cdot c_{52}(\epsilon,a,b,c_{50},c_{51})}{n}$$

whenever $\vartheta_k^{(n)}$ and $\vartheta_{k+1}^{(n)}$ lie in $[\alpha+\epsilon,\beta-\epsilon]$, $\epsilon$ being any small positive number.

Note on the printed lower bound. With the theorem's labels, $b=\cos\beta<a=\cos\alpha$, so the printed $b-a$ is negative. The introduction's version (22) (p. 518) names the endpoints the other way round ($a=\cos\beta$, $b=\cos\alpha$) and prints the same $[\epsilon(b-a)]^{1/2}$, which there is real; the proof (p. 536) applies the Bernstein--Fejér inequality on the subinterval. The lower bound is therefore read with the length $a-b$ of the subinterval under the root. The proof of the upper bound also uses $c_{49}\ge1$ (p. 537).

Proof pointer

Pp. 536--537. The lower bound: $l_k(\cos\vartheta)$ is a trigonometric polynomial of order $n-1$ taking the values $1$ and $0$ at $\vartheta_k^{(n)}$ and $\vartheta_{k+1}^{(n)}$, so the mean-value theorem and the Bernstein--Fejér inequality bound $1/(\vartheta_{k+1}-\vartheta_k)$. The upper bound: if the largest gap in $[\alpha+\epsilon,\beta-\epsilon]$ is $2A(n)/n$, the paper interpolates the non-negative cosine polynomial (47), a sum of two powers of Fejér-type kernels centered at the gap's midpoint $\delta_0$, at the nodes; using the lower bound already proved to space the other nodes, (49) yields $1<c/n^2+c'/A^2$, so $A(n)$ is bounded.

Read depth

Claims checked: Theorem VII and the introduction's (22) were read clause by clause on the page images of the print; the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input named by the paper: the Bernstein--Fejér inequality for derivatives of polynomials.

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.