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Statement

Theorem X (p. 543). In $[-1,1]$ let $p(x)\sqrt{1-x^2}$ be continuous with $p(x)\sqrt{1-x^2}\ge m>0$. Then for every $\epsilon>0$, $n>n_0(\epsilon)$ and every node with $|x_\nu^{(n)}|\le[1-\log n/n^2]^{1/2}$, uniformly for $-1\le x\le1$,

$$|l_{\nu,n}(x)-\phi_{n-1}(x)| =\left|l_\nu(x)- \frac{T_{n-1}(x_\nu^{(n)})T_n(x)-T_n(x_\nu^{(n)})T_{n-1}(x)} {\Bigl(n-\frac12+\frac12\frac{\sin(2n-1)\vartheta_\nu^{(n)}}{\sin\vartheta_\nu^{(n)}}\Bigr)(x-x_\nu^{(n)})}\right| <\epsilon,$$

with $x_\nu^{(n)}=\cos\vartheta_\nu^{(n)}$ and the Chebyshev polynomials $T_r(\cos\vartheta)=\cos r\vartheta$; the theorem prints "$r>1$" in this parenthesis, while the definition (52) (p. 539) sets $T_0(x)=1/\sqrt2$ and $T_r(\cos\vartheta)=\cos r\vartheta$ for $r\ge1$. Here $\phi_{n-1}$ is the polynomial (52) with $\xi_0=x_\nu^{(n)}$.

Remarks (pp. 544--545). Remark I says that if instead $p(x)$ itself is continuous and at least $m$ on a subinterval $[a,b]$, Theorems IX and X remain true for the nodes and points in $[a+\epsilon,b-\epsilon]$, with Legendre polynomials replacing the Chebyshev polynomials in $\phi_{n-1}$. Remark II calls it probable that Theorem X holds for the fundamental functions of every node; this is not proved.

Proof pointer

Pp. 543--544. With $I_\nu=\int_{-1}^1(l_\nu-\phi_{n-1})^2p$, identity (66) gives $I_\nu=-k_\nu+\int_{-1}^1\phi_{n-1}^2p$, so Theorem IX and Lemma IX give $nI_\nu\to0$ uniformly in $\nu$ (67). Remark I to Theorem VIII and (55) bound $l_\nu-\phi_{n-1}$, the Bernstein--Fejér inequality bounds its derivative by a constant times $n$, so near a point where $|l_\nu-\phi_{n-1}|$ reaches its maximum $D_\nu$ it stays above $D_\nu/2$ on an interval of length of order $D_\nu/n$; this bounds $nI_\nu$ from below by a positive function of $D_\nu$, and (67) gives $D_\nu\to0$.

Read depth

Claims checked: Theorem X, (52) and Remarks I and II 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 IX, Lemma IX and Remark I to Theorem VIII of the same paper; the Bernstein--Fejér inequality.

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.