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Statement

Theorem XVI (p. 553). Let the weight $p$ be $L$-integrable with $p(x)\ge m>0$ in $[-1,1]$, and let the roots of the $n$th orthogonal polynomial be $\cos\vartheta_\nu^{(n)}$. Then for every $[\alpha,\beta]$ in $[0,\pi]$

$$\Bigl|\sum_{\alpha\le\vartheta_\nu^{(n)}\le\beta}1-\frac{\beta-\alpha}{\pi}n\Bigr| <c_{90}(p,\epsilon)\,n^{1/2+\epsilon}.$$

Proof pointer

P. 553. By (50) (on the Theorem VIII page) taken over all of $[-1,1]$, the fundamental functions are bounded by a constant depending on $m$ and $\int_{-1}^1p$ times $n$, so Theorem XV applies.

Read depth

Claims checked: Theorem XVI 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 XV and the bound (50) 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.