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Statement

Theorem IV (p. 530). Add to the hypotheses of Theorem III that, throughout a subinterval $[c,d]$ of $[a,b]$,

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

If $x_d^{(n)}$ again denotes the root of $\omega_n(x)$ nearest to $x$, then for $c+\epsilon\le x\le d-\epsilon$ and $n>n_0(\epsilon,c,d,p)$

$$|\omega_n(x)|>\frac{c_{37}}{\sqrt{b-a}} \left[\frac{m}{M+\int_{-1}^1p(t)\,dt}\right]^{1/2} |x-x_d^{(n)}|\left(\frac{b-a}{4}\right)^n\sqrt n.$$

Remark I (p. 531). In the special case $m\le p(x)\le M/\sqrt{1-x^2}$ throughout $[-1,1]$, the paper records that on $[-1+\epsilon,1-\epsilon]$

$$c_{41}(p,\epsilon)\frac{\sqrt n}{2^n}|x-x_d^{(n)}|\le|\omega_n(x)| \le c_{42}(p,\epsilon)\frac{\sqrt n}{2^n}.$$

Proof pointer

Pp. 530--531. Lemma V (p. 530): if $p\ge0$ is $L$-integrable on $[-1,1]$ and $p(x)\le M/\sqrt{1-x^2}$ on $[u,v]$, then the Christoffel numbers of the nodes in $[u+\eta,v-\eta]$ ($\eta>0$) are $O\bigl((M+\eta^{-2}n^{-1}\int_{-1}^1p)/n\bigr)$, with numerical constants; it is proved from Shohat's minimum property with a Fejér-kernel test polynomial. The proof of Theorem IV inserts Lemma V into identity (39) with $u=c$, $v=d$, $\eta=\frac12\epsilon$, uses footnote 7 to place the root interval containing $x$ inside $[c+\frac12\epsilon,d-\frac12\epsilon]$ for large $n$, and applies Lemma IV.

Read depth

Claims checked: Theorem IV, Lemma V 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 III and its identity (39), Lemma IV and Lemma V of the same paper; Fejér's density result of footnote 7.

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.