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Statement

Corollaire (printed pp. 1049--1050). Let $n\ge1$ and let $2n+1$ distinct points $z_0,\ldots,z_{2n}$ be given on the unit circle. Among the polynomials $P_{2n}(z)$ of degree $2n$ whose modulus is at most $1$ at these points, some reach modulus $(\frac2\pi-o(1))\log n$ on the circle as $n\to\infty$, whatever the points.

The printed hypothesis reads "atteint la valeur 1" at the points; the proof on p. 1050 takes the polynomial that realizes the largest modulus at a point $z$, and the value it displays is that largest modulus under the bound $|P_{2n}(z_k)|\le1$ at $z_0,\ldots,z_{2n}$. The statement above is given in that reading. With $A_{2n+1}(z)=(z-z_0)\cdots(z-z_{2n})$ that largest modulus is

$$|A_{2n+1}(z)|\sum_{k=0}^{2n}\frac1{|z-z_k|\,|A_{2n+1}'(z_k)|},$$

and for $z=e^{i\varphi}$, $z_k=e^{i\theta_k}$ the paper identifies it with the trigonometric Lebesgue function (37 bis) of the points $\theta_k$, so the corollary follows from the Théorème.

Source. Serge Bernstein, Sur la limitation des valeurs d'un polynôme $P_n(x)$ de degré $n$ sur tout un segment par ses valeurs en $(n+1)$ points du segment, Bull. Acad. Sci. URSS, Classe des sciences mathématiques et naturelles, VII série (1931), no. 8, 1025--1050; the Corollaire and its proof on printed pp. 1049--1050 (PDF pp. 25--26). The paper writes "la circonférence $c$"; its proof takes $z_k=e^{i\theta_k}$. The copy read is identified on the source card.

Read depth. Claims checked: the statement and its reduction to (37 bis) were read on the page images. The reduction is an identity between moduli ($|e^{i\varphi}-e^{i\theta}|=2|\sin\frac{\varphi-\theta}2|$); the Théorème it rests on is claims-checked only.

Proof pointer

Page 1050: the paper states that the extremal value at $z$ is the displayed sum (the interpolation formula with node values of modulus one chosen to align every term, a step it does not write out); writing the factors through $|e^{i\varphi}-e^{i\theta}|$ turns it into (37 bis), and the Théorème applies.

Dependencies

The Théorème of p. 1041, with its interfaces listed there.

Bears on

• Problem 1129, its adjacent unit-circle variant (Erdős's question on nodes on the circle, recorded on that page): for an odd number $2n+1$ of nodes the corollary gives the asymptotic lower bound $(\frac2\pi-o(1))\log n$ for the maximum of the circle Lebesgue function. It says nothing about which nodes minimize, and the variant carries no claim page there.