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Statement

Setting (p. 263, Nr. 1). For real nodes $-1\le x_n<x_{n-1}<\cdots<x_2<x_1\le1$, $l_k(x)$ is the Lagrange fundamental function of the node $x_k$: the polynomial of degree exactly $n-1$ equal to $1$ at $x_k$ and $0$ at the other nodes. $P_m$ denotes the $m$-th Legendre polynomial.

Main theorem (pp. 264--265, formulas (6)--(9'); the paper gives it no number and calls it a theorem of Chebyshev type, p. 265).

Formula (6): over all node sets $-1\le x_n<\cdots<x_1\le1$, the least value of $\max_{-1\le x\le1}\{(l_1(x))^2+\cdots+(l_n(x))^2\}$ is $1$:

$$\min_{-1\le x_n<x_{n-1}<\cdots<x_1\le1}\ \max_{-1\le x\le1}\bigl\{(l_1(x))^2+\cdots+(l_n(x))^2\bigr\}=1.$$

Formulas (7)--(8): for $n\ge2$ the only node set $x_1,\ldots,x_n$ with $\max_{-1\le x\le1}\{(l_1(x))^2+\cdots+(l_n(x))^2\}=1$ is the set of the $n$ roots of

$$(1-x^2)\,P'_{n-1}(x)=0,$$

where $P'_{n-1}$ is the derivative in $x$ of the $(n-1)$-st Legendre polynomial. These are the endpoints $\pm1$ and the $n-2$ zeros of $P'_{n-1}$.

Formulas (9) and (9'): for this node set and every value of $x$,

$$(l_1(x))^2+\cdots+(l_n(x))^2=1-\frac{(1-x^2)\,(P'_{n-1}(x))^2}{n(n-1)}\qquad(n=2,3,\ldots),$$

or, with $x=\cos\theta$, the sum equals $1-\frac{1}{n(n-1)}\bigl(\frac{dP_{n-1}}{d\theta}\bigr)^2$.

Equivalent descriptions of the nodes (p. 267, (27)--(29)): the roots of $\int_{-1}^xP_{n-1}(t)\,dt=0$, of $(1-x^2)P'_{n-1}(x)=0$, or of $P_n(x)-P_{n-2}(x)=0$. Footnote 6 (p. 267) identifies them as the zeros of the paper's Jacobi polynomial $J_n(0,0,x)$, in a parametrization where $J_n(\alpha,\beta,x)$ satisfies $(1-x^2)\omega''+[2(\alpha-\beta)-2(\alpha+\beta)x]\omega'+n[n+2(\alpha+\beta)-1]\omega=0$, and records $J_n(0,0,x)=(1-x^2)P'_{n-1}(x)=-n(n-1)\int_{-1}^xP_{n-1}(t)\,dt=-\frac{n(n-1)}{2n-1}\bigl(P_n(x)-P_{n-2}(x)\bigr)$.

Further forms of the sum (p. 269, (45) and (47)), for the same nodes:

$$\sum_{k=1}^n(l_k(x))^2=1-\frac{n(n-1)}{1-x^2}\Bigl(\int_{-1}^xP_{n-1}(t)\,dt\Bigr)^2 =1-\frac{n(n-1)}{(2n-1)^2}\,\frac{1}{1-x^2}\bigl(P_n(x)-P_{n-2}(x)\bigr)^2 .$$

Limit (p. 270, (49)), stated as following easily from (45)--(48): for these nodes, $\lim_{n\to\infty}\sum_{k=1}^n(l_k(x))^2=1$ for $-1\le x\le1$, uniformly on every interval $-1+\varepsilon\le x\le1-\varepsilon$ with $\varepsilon>0$, and not uniformly on the whole interval $-1\le x\le1$.

Proof pointer

Pp. 265--269 (Nr. 3--6). The sum equals $1$ at every node, which gives the lower bound (11). If the sum is at most $1$ on $[-1,1]$, then $|l_k(x)|\le1$ there, so each interior node is a maximum point of its own $l_k$ and $l_k'(x_k)=0$. With $\omega(x)=\prod_k(x-x_k)$ this reads $\omega''(x_k)=0$ at the interior nodes, which forces $x_1=1$, $x_n=-1$ and the differential equation $(1-x^2)\omega''+n(n-1)\omega=0$; its polynomial solution vanishing at $-1$ is a multiple of $\int_{-1}^xP_{n-1}$ (pp. 265--267). Conversely, setting all values to $1$ in Hermite's step-parabola interpolation gives the identity $\sum_kv_k(x)(l_k(x))^2\equiv1$ with $v_k(x)=1-\frac{\omega''(x_k)}{\omega'(x_k)}(x-x_k)$ (p. 268, (33)--(34)). For these nodes $v_k\equiv1$ for $2\le k\le n-1$, while $v_1(x)=1+\frac{n(n-1)}{2}(1-x)$ and $v_n(x)=1+\frac{n(n-1)}{2}(1+x)$ are at least $1$ on $[-1,1]$, so the sum is at most $1$ there (p. 268, (39)--(40)). The same identity yields the closed forms (45)--(48) (p. 269).

Read depth

Claims checked: the statement, (6)--(9'), (27)--(29), footnote 6, (45), (47) and (49) were read clause by clause on the page images of the print, and the proof of Nr. 3--6 was followed. The paper gives no proof of (49). 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: Legendre's differential equation and Hermite's step-parabola interpolation formula, (30)--(32), both standard.

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: the theorem minimizes the maximum of $\sum_k(l_k(x))^2$ on $[-1,1]$, not the integral $I$ the problem asks about, and the paper says nothing about the least value of $I$. Its extremal nodes are the roots of the integral of the Legendre polynomial that the problem page names. Integrating (9) with $\int_{-1}^1(1-x^2)(P'_m(x))^2\,dx=\frac{2m(m+1)}{2m+1}$ at $m=n-1$ gives $I=2-\frac{2}{2n-1}$ for these nodes, the upper bound the problem page records from Erdős, Szabados, Varma and Vértesi; that integration is made here, not in the paper.