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Statement

Theorem VIII (p. 538). Let the weight $p(x)$ be non-negative and $L$-integrable throughout $[-1,1]$, and suppose that throughout the subinterval $[b,a]\equiv[\cos\beta,\cos\alpha]$

$$0<m\le p(x)\le\frac{M}{\sqrt{1-x^2}}.$$

Then for every $\epsilon>0$, the roots $\cos\vartheta_k^{(n)}$ of the $n$th orthogonal polynomial satisfy

$$\frac{c_{56}(a,b,p,\epsilon)}{n}\le\vartheta_{k+1}^{(n)}-\vartheta_k^{(n)} \le\frac{c_{57}(a,b,p,\epsilon)}{n}$$

if $\alpha+\epsilon\le\vartheta_k^{(n)}<\vartheta_{k+1}^{(n)}\le\beta-\epsilon$.

Remark I (p. 538). If $m/\sqrt{1-x^2}\le p(x)\le M/\sqrt{1-x^2}$ on $[-1,1]$, then $\sum_\nu l_\nu(x)^2\le c_{60}M/m$. The paper uses this bound in the proofs of Theorems X and XVII.

Proof pointer

P. 538. The corollary (34a) of Lemma II and $k_\nu<\int_{-1}^1p$ give (50), $|l_\nu(x)|<[\frac{2}{m(a-b)}\int_{-1}^1p]^{1/2}n$ on $[b,a]$; (34b) with Lemma V (stated on the Theorem IV page) bounds the fundamental functions of the nodes in $[b+\frac12\epsilon,a-\frac12\epsilon]$ by a constant there. The hypotheses of Theorem VII then hold on that smaller interval with $c_{51}=1$.

Read depth

Claims checked: Theorem VIII, (50) and Remark I were read clause by clause on the page images of the print; the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

Theorem VII, Lemmas II and V and the corollaries (34a), (34b) 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.