Fejer 1932 bestimmung derjenigen abszissen eines intervalles

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equation_59: Fejér's closed form for the sum of squares of the Lagrange fundamental functions at the n Chebyshev nodes cos((2k+1)pi/(2n)), with its consequences that the sum is at most 2 - 1/n on [-1,1] and tends to 1 inside the interval and to 2 at the endpoints.

equation_69: Fejér's identity that for Lagrange trigonometric interpolation at the 2n+1 equally spaced nodes 2k pi/(2n+1) the squares of the fundamental trigonometric polynomials of order n sum to 1 identically.

equation_97: Fejér's limit theorem that for the zeros of the n-th Legendre polynomial the sum of squares of the Lagrange fundamental functions tends to 1 at every point of (-1,1) and to plus infinity at x = 1 and x = -1 as n grows.

main_theorem: Fejér's theorem that, over nodes in [-1,1], the least possible maximum on [-1,1] of the sum of squares of the Lagrange fundamental functions is 1, that for n >= 2 it is attained only at the n roots of (1-x^2)P'{n-1}(x), and that for these nodes the sum equals 1 - (1-x^2)P'{n-1}(x)^2/(n(n-1)) for every x.

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Fejér, Leopold, Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein Möglichst kleines Maximum Besitzt. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1(3) (1932), 263--276. The file's Numdam cover page prints "© Scuola Normale Superiore, Pisa, 1932, tous droits réservés." and "Toute copie ou impression de ce fichier doit contenir la présente mention de copyright." and refers to the Numdam conditions of use (http://www.numdam.org/conditions), every other right reserved.

This German-language paper solves a Chebyshev-type extremal problem for Lagrange interpolation on $[-1,1]$. For $n$ distinct nodes $-1\le x_n<\cdots<x_1\le1$ with fundamental functions $l_k(x)$, Fejér considers the sum of squares $\sum_k(l_k(x))^2$. Its maximum over the interval is always at least $1$, since the sum equals $1$ at each node. The main theorem (pp. 264--265, formulas (6)--(9')) says that this bound is the least possible maximum and that, for $n\ge2$, exactly one node set attains it: the $n$ roots of $(1-x^2)P'_{n-1}(x)$, that is, the endpoints $\pm1$ and the $n-2$ zeros of the derivative of the $(n-1)$-st Legendre polynomial, equivalently the roots of $\int_{-1}^xP_{n-1}(t)\,dt$ (p. 267, (27)--(29)). For these nodes the sum equals $1-(1-x^2)(P'_{n-1}(x))^2/(n(n-1))$ for every $x$ (p. 265, (9)). The proof (pp. 265--269) starts from the assumption that the sum is at most $1$ on $[-1,1]$: then $|l_k(x)|\le1$ there, each interior node is a maximum point of its own $l_k$, and the resulting conditions $l_k'(x_k)=0$ determine the nodes through a differential equation. The converse follows from an identity obtained from Hermite's step-parabola interpolation (p. 268, (33)).

Section 1 opens (pp. 263--264) with the analogous problem for the sum of the $|l_k(x)|$, whose least maximum $M_n$ and extremal nodes the paper calls unknown, recording $\frac1{12}\log n<M_n<12\log n$ ($n=2,3,\ldots$; (4), the lower bound credited to Faber). Footnote 4 (pp. 264--265) motivates both extremum problems through Tietze's table error $T(x)=\varepsilon_1l_1(x)+\cdots+\varepsilon_nl_n(x)$, bounded by $\varepsilon\sum_k|l_k(x)|$ with $\varepsilon=\max_k|\varepsilon_k|$ and by $\sqrt{\varepsilon_1^2+\cdots+\varepsilon_n^2}\sqrt{\sum_k(l_k(x))^2}$. Section 2 computes the sum in closed form at the Chebyshev nodes (p. 271, (58)--(61)) and shows it is identically $1$ for classical trigonometric interpolation at $2n+1$ equally spaced nodes (p. 272, (69)). Section 3 turns to the zeros of the paper's Jacobi polynomials $J_n(\alpha,\beta,x)$ with $0\le\alpha<\frac12$, $0\le\beta<\frac12$: it recalls from Fejér's Math. Ann. 106 paper the bound $\max(\frac1{1-2\alpha},\frac1{1-2\beta})$ for the sum on $[-1,1]$ ((75), p. 273), announces without proof the limit (76) (p. 273), equal to $\frac1{1-2\beta}$ at $x=1$, $1$ for $-1<x<1$ and $\frac1{1-2\alpha}$ at $x=-1$, and proves for the Legendre--Gauss nodes that the sum tends to $1$ inside $(-1,1)$ and to $+\infty$ at $\pm1$ (pp. 274--276, (97)).

Source: http://www.numdam.org/item/ASNSP_1932_2_1_3_263_0/.

Read status: claims checked for the main theorem with (27)--(29), (45), (47) and (49), and for (59)--(61), (69) and (97), read clause by clause on the page images of the print; the proofs of the main theorem, (59), (69) and (97) were followed, the last relying on cited convergence theorems of Stieltjes and of Fejér (1916). (75) and (76) are recalled or announced without proof and were not checked. A second reader checked the result pages' statements, hypotheses, labels and pages against the print; the proofs were not independently reviewed. Result pages: main_theorem, equation_59, equation_69 and equation_97.

Bears on. #1131: the paper minimizes the maximum of $\sum_k(l_k(x))^2$ on $[-1,1]$, not the integral $I$ the problem asks about, and says nothing about the least value of $I$. Its main theorem identifies the extremal nodes for the maximum as the roots of the integral of the Legendre polynomial, the nodes the problem page names, and its formula (9) gives the integrand of $I$ at those nodes in closed form; integrating it gives the value $2-\frac{2}{2n-1}$ that the problem page records as the upper bound of Erdős, Szabados, Varma and Vértesi, a computation made on the result page, not in the paper. Formula (59) and formula (97) give the same integrand in closed form at the Chebyshev nodes and its pointwise limit as $n\to\infty$ at the Legendre--Gauss nodes.

Results.

• Main theorem (pp. 264--265, (6)--(9')): the least maximum on $[-1,1]$ of $\sum_k(l_k(x))^2$ is $1$, attained for $n\ge2$ only at the roots of $(1-x^2)P'_{n-1}(x)$, where the sum equals $1-(1-x^2)(P'_{n-1}(x))^2/(n(n-1))$.
• Formulas (58)--(61) (p. 271): at the Chebyshev nodes the sum equals $1-\frac1{2n}+\frac1{2n}\frac{\sin(2n-1)\theta}{\sin\theta}$, is at most $2-\frac1n$, and tends to $1$ inside and to $2$ at $\pm1$.
• Formula (69) (p. 272): for trigonometric interpolation at $2n+1$ equally spaced nodes the sum of squares of the fundamental polynomials is identically $1$.
• Formula (97) (p. 276): at the Legendre--Gauss nodes the sum tends to $1$ on $(-1,1)$ and to $+\infty$ at $x=\pm1$.

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