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Statement

Setting (p. 274, Nr. 11). The nodes $x_1,\ldots,x_n$ are the roots of $P_n(x)=0$, where $P_n$ is the $n$-th Legendre polynomial (the Legendre--Gauss nodes; the paper's limiting case $\alpha=\beta=\frac12$ of its Jacobi nodes), and $l_k(x)$ are the Lagrange fundamental functions.

Formula (97) (p. 276, stated as the result of Nr. 11--12). For these nodes,

$$\lim_{n\to\infty}\bigl\{(l_1(x))^2+(l_2(x))^2+\cdots+(l_n(x))^2\bigr\}= \begin{cases}+\infty,&x=1,\\ 1,&-1<x<1,\\ +\infty,&x=-1.\end{cases}$$

The endpoint part is (88) (p. 275) and the interior part is (96) (p. 276).

Proof pointer

Pp. 274--276. At $x=\pm1$: with $l_k(1)^2=1/((1-x_k)^2(P_n'(x_k))^2)$, the Gauss quadrature sum $Q_n(f)$ of the function equal to $\frac12\frac{1+x}{1-x}$ on $[-1+\varepsilon,1-\varepsilon]$ and $0$ elsewhere is at most $\sum_kl_k(1)^2$ ((81)--(86)); by Stieltjes's convergence theorem $Q_n(f)$ tends to $\log\frac{2-\varepsilon}{\varepsilon}-(1-\varepsilon)$ ((87)), which is unbounded as $\varepsilon\to0$. At an interior point $a$: for $f(x)=\frac{1-x^2}{1-2ax+x^2}$ the $n$-th Hermite step parabola at the Legendre--Gauss nodes evaluated at $a$ equals $\sum_k(l_k(a))^2$ ((91)--(94)), and Fejér's earlier theorem on step parabolas (90), cited as Theorem VII of his 1916 Göttingen paper, gives the limit $f(a)=1$ ((95)--(96)).

Read depth

Claims checked: (77), (88), (96) and (97) were read on the page images of the print and the argument of Nr. 11--12 was followed. The convergence theorems of Stieltjes and of Fejér's 1916 paper are cited, not proved here. A second reader checked the statement, hypotheses, label and page against the print; the proof was not independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Gauss quadrature with Stieltjes's convergence theorem, and Fejér, Über Interpolation, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1916, 66--91, Theorem VII.

Source. L. Fejér, Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein Möglichst kleines Maximum Besitzt, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 3, 263--276; the edition read is named on the source card.

Bears on

• Problem 1131: (97) gives the pointwise limit as $n\to\infty$ of the integrand of the problem's $I$ at the Legendre--Gauss nodes; the paper does not integrate it and says nothing about the least value of $I$.