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Statement

Conjecture (printed pp. 1026--1027, unnumbered). Take $n+1$ nodes $a_0<\cdots<a_n$ in the segment $[-1,1]$, nodal polynomial $A_{n+1}(x)=(x-a_0)\cdots(x-a_n)$, and the Lebesgue function $F$ of equation (1),

$$F(x)=\sum_{i=0}^{n}\left|\frac{A_{n+1}(x)}{(x-a_i)A_{n+1}'(a_i)}\right|.$$

$F$ has one maximum on each of the $n+2$ intervals $(-1,a_0),(a_0,a_1),\ldots,(a_n,1)$; on each it is the largest absolute value there of a polynomial of degree $n$ bounded by $1$ in absolute value at the nodes (p. 1026). Bernstein writes that it seems probable that the largest of these $n+2$ maxima is minimized when all the maxima are equal. He adds that he could prove this only under the condition that $n$ grows indefinitely, and that the paper treats only that case.

Footnoted example (printed p. 1027). For $n=2$ the footnote says that the polynomial printed as $x^2-\frac89$ leads to the minimum $M=\frac54$ of the maxima of $F$, attained at the four points $\pm1$ and $\pm\frac{\sqrt2}3$, and that the Chebyshev polynomial $x(x^2-\frac34)$, which gives $M=\frac53$, therefore does not realize the minimum for every $n$. A nodal polynomial for $n=2$ has degree three, so the printed $x^2-\frac89$ lacks a factor; the nodes meant are $0,\pm\frac{2\sqrt2}3$, the zeros of $x(x^2-\frac89)$. For these nodes a direct computation, made for this page, gives the maximum $\frac54$ at the four stated points, and $\frac53$ for the Chebyshev nodes $0,\pm\frac{\sqrt3}2$.

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 conjecture on printed pp. 1026--1027 and its footnote on p. 1027 (PDF pp. 2--3). The question it refines is posed on p. 1025, recalled from the author's note in Comm. Soc. Math. Charkow, t. XIV (1914). The copy read is identified on the source card.

Read depth. Claims checked: the conjecture and the footnote were read clause by clause on the page images. The paper gives no proof of the conjecture; its later sections treat only the case of large $n$.

Bears on

• Problem 1129: this conjectures an equal-maxima characterization of the minimizing nodes, the kind of description the problem asks for, as Bernstein posed it, with the nodes free in the segment and $n+2$ maxima counting the two end intervals. The problem page records Erdős's form of the conjecture, the de Boor--Pinkus theorem for node systems containing both endpoints, and Kilgore's independent proof; this page does not say whether the free-node form, as Bernstein printed it, is settled. The footnoted three-node minimum $\frac54$ is the value that page cites from p. 1027.