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Statement

Setting (pp. 510--512). The $n$th row of the node matrix $\mathfrak M$ is $1\ge x_1^{(n)}>x_2^{(n)}>\cdots>x_n^{(n)}\ge-1$ (display (2)), $\omega_n(x)=\prod_{\nu=1}^n(x-x_\nu^{(n)})$ (display (5b)) and $l_\nu(x)=\omega_n(x)/\bigl(\omega_n'(x_\nu)(x-x_\nu)\bigr)$ are the fundamental functions of Lagrange interpolation. Following Fejér, the matrix is strongly normal (display (9b), p. 512) when there is a positive constant $c_1$, independent of $x$, $n$ and $k$, with

$$1-\frac{\omega_n''(x_k)}{\omega_n'(x_k)}(x-x_k)\ge c_1 \qquad(-1\le x\le1,\ k=1,\ldots,n,\ n=1,2,\ldots);$$

then $|l_k(x)|\le1/\sqrt{c_1}$ on $[-1,1]$ (display (11)).

Theorem I (p. 521, quoted). "For strongly normal matrices we have in $[-1,+1]$

$$|\omega_n(x)|\le\frac{8}{\sqrt{c_1}}\cdot\frac{\sqrt n}{2^n}, \qquad n=1,2,\cdots."$$

The paper says (p. 522) that the bound cannot be essentially improved on $[-1,1]$, pointing to the strongly normal matrix of roots of the Jacobi polynomials with both parameters equal to $-\epsilon$, whose value at $1$ it gives as $\sim c_{31}(\epsilon)n^{1/2-\epsilon}/2^n$. In the introduction (p. 515) it calls it probable that on $[-1+\epsilon,1-\epsilon]$ the factor $\sqrt n$ can be omitted, with $8/\sqrt{c_1}$ replaced by a constant $c_{10}(c_1,\epsilon)$; this is not proved.

Proof pointer

Pp. 521--522. The paper gives two proofs. The first compares the arithmetic and geometric means of $l_\nu(x)^2$ and uses Schur's bound for $\prod_\nu|\omega_n'(x_\nu)|$ over nodes in $[-1,1]$; it yields the same shape with a constant $c_{30}$. The second, which gives the constant $8/\sqrt{c_1}$, rests on Lemma I (p. 521): if $1\ge x_1^{(n)}>\cdots>x_n^{(n)}\ge-1$ then $\sum_{\nu=1}^n1/|\omega_n'(x_\nu)|\ge2^{n-2}$, with equality only for $\omega_n(x)=(x^2-1)U_{n-2}(x)$, where $U_k(\cos\vartheta)=2^{-k}\sin(k+1)\vartheta/\sin\vartheta$. Lemma I is proved through the extremal polynomial of degree $n-1$ with leading coefficient $1$ that minimizes the maximum of its absolute values at the nodes, compared with the Chebyshev polynomial. Writing $|\omega_n(x)|\sum_\nu1/|\omega_n'(x_\nu)|\le2\sum_\nu|l_\nu(x)|$ and applying Cauchy--Schwarz with $\sum_\nu l_\nu(x)^2\le1/c_1$ (from Fejér's identity (10)) gives the theorem.

Read depth

Claims checked: the definitions, Theorem I and Lemma I were read clause by clause on the page images of the print; both proofs were followed for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Fejér's definition of strongly normal matrices and identity (10), and Schur's theorem on the product $\prod|\omega_n'(x_\nu)|$ (for the first proof only).

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.