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Statement

Theorem XVII (p. 553). Let the weight $p(x)$ be $L$-integrable with $0<m\le p(x)\sqrt{1-x^2}\le M$ in $[-1,1]$. Then for the roots $\cos\vartheta_\nu^{(n)}$ of the $n$th orthogonal polynomial and every subinterval $[\alpha,\beta]$ of $[0,\pi]$

$$\Bigl|\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1-\frac{\beta-\alpha}{\pi}n\Bigr| <c_{91}(p,\epsilon)\{(\beta-\alpha)n\}^{1/2+\epsilon}$$

if $n(\beta-\alpha)>c_{92}(p,\epsilon)$.

The introduction (pp. 519--520) states this as (23) under $M\ge p(x)\sqrt{1-x^2}\ge m$ on $[-1,1]$.

Proof pointer

P. 553. Remark I to Theorem VIII bounds the fundamental functions uniformly by a constant depending on $M/m$, so Theorem XIV applies.

Read depth

Claims checked: Theorem XVII and the bound on p. 553 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Theorem XIV and Remark I to Theorem VIII of the same 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.