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Statement

Setting (p. 272, Nr. 9). The interval $0\le\theta<2\pi$ is divided into $2n+1$ equal parts, with nodes

$$\theta_k=k\,\frac{2\pi}{2n+1},\qquad k=0,1,2,\ldots,2n,$$

and $\lambda_k(\theta)$ is the $k$-th fundamental polynomial of the classical Lagrange trigonometric interpolation at these nodes, a trigonometric polynomial of order $n$ equal to $1$ at $\theta_k$ and $0$ at the other nodes.

Formula (69) (p. 272, stated as the result of Nr. 9). For these nodes,

$$(\lambda_0(\theta))^2+(\lambda_1(\theta))^2+\cdots+(\lambda_{2n}(\theta))^2\equiv1 .$$

The paper sets this beside the known identity (70), $\lambda_0(\theta)+\cdots+\lambda_{2n}(\theta)\equiv1$ (p. 273).

Proof pointer

P. 272, (62)--(68). The matrix of the values at $\theta_k$ of the $2n+1$ normalized functions $\sqrt{1/(2n+1)}$, $\sqrt{2/(2n+1)}\cos v\theta$, $\sqrt{2/(2n+1)}\sin v\theta$ ($1\le v\le n$) is orthogonal, so the $\lambda_k$ are an orthogonal transformation of these functions, and the sum of squares equals $\frac1{2n+1}+\frac{2n}{2n+1}=1$.

Read depth

Claims checked: (62), (69) and (70) were read on the page images of the print and the derivation (63)--(68) was followed. A second reader checked the statement, hypotheses, label and page against the print; the proof was not independently reviewed.

Dependencies

None in the corpus.

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

None recorded: the identity concerns trigonometric interpolation on the circle, not the algebraic interpolation on $[-1,1]$ of Problem 1131.