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Statement

Théorème (printed p. 1041). Let $n\ge1$ and let $2n+1$ distinct points $\theta_0,\ldots,\theta_{2n}$ be given in $(-\pi,\pi)$. Consider the trigonometric sums of order $n$, with real or complex coefficients,

$$S_n(\theta)=A_0+A_1\cos\theta+B_1\sin\theta+\cdots +A_n\cos n\theta+B_n\sin n\theta ,$$

whose modulus is at most $1$ at each of the given points. Whatever the points, such a sum "pourra atteindre asymptotiquement" the value $\frac2\pi\log n$: the largest modulus attainable by these sums is at least $(\frac2\pi-o(1))\log n$ as $n\to\infty$. Since the points are arbitrary for each $n$, the $o(1)$ does not depend on them.

The quantity bounded is, at each $\theta$, the trigonometric Lebesgue function of the points, equation (37) on printed p. 1042: with $Q_{n+1}(\theta)=\sin\frac{\theta-\theta_0}2\cdots\sin\frac{\theta-\theta_{2n}}2$ (the paper normalizes $\theta_0=0$ in (36)),

$$F_n(\theta)=\left|\frac{Q_{n+1}(\theta)}2\right| \sum_{i=0}^{2n}\frac1{\left|\sin\frac{\theta-\theta_i}2\,Q_{n+1}'(\theta_i)\right|},$$

the largest modulus at $\theta$ of an order-$n$ sum bounded by $1$ at the points; the extremal sums have real coefficients, or coefficients of one common argument (p. 1043). The proof ends with (58) on printed p. 1049, $F_n(\varphi)\gtrsim\frac2\pi\log n$, at a point $\varphi$ where $|Q_{n+1}|$ attains its maximum.

Algebraic consequence (printed p. 1042). The paper then states that for any choice of $n+1$ points $a_i=\cos\theta_i$ ($i=0,\ldots,n$) of the segment $(-1,+1)$, the polynomial of degree $n$ bounded by $1$ in absolute value at these points can reach the value $\frac2\pi\log n$ asymptotically. In the notation of the source card, the maximum over the segment of the Lebesgue function $F$ of equation (1) is at least $(\frac2\pi-o(1))\log n$; with the equal-maxima class of section 3 this is the asymptotic $M\sim\frac2\pi\log n$ announced as equation (2) on printed p. 1026. The step on p. 1042 takes sums with $|S_n(\theta_i)|\le1$ for $i=0,\ldots,n$ and $S_n(\theta_i)=S_n(-\theta_i)$ for $i=1,\ldots,n$, argues that such a sum contains no sines, and substitutes $x=\cos\theta$.

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 Théorème on printed p. 1041, the algebraic consequence on p. 1042, the proof on pp. 1042--1049 (PDF pp. 17--25 of the 26-page scan). The copy read is identified on the source card.

Read depth. Claims checked: the Théorème, the algebraic consequence and the final display (58) were read clause by clause on the page images. The proof was read for its structure and not checked; this page reconstructs neither the trigonometric proof nor the algebraic transfer.

Proof pointer

Pages 1042--1049. The trigonometric interpolation formula (35)--(37) gives $F_n$. For consecutive points the paper compares the midpoint quantity $H_i$ of (38)--(39), the analogue of the algebraic inequality (27), with a simpler product $I_i$ through (40)--(42), and shows on pp. 1044--1045, (43)--(45), that $I_i$ increases in its own gap, decreases in the others and is convex in each gap, so that a symmetric function of the $I_i$ with nonnegative successive derivatives is smallest for equal gaps $\delta_i=\pi/(2n+1)$. Over arcs of length $\alpha=n^{-(1-\varepsilon)}$ it truncates the products, (47)--(52), bounds the error using the gap bound (28) and a lower bound (53) on the gaps obtained from a derivative estimate on p. 1048, and reaches the essential inequality (55), $S_\alpha\gtrsim\alpha/(2\pi)$. Summing these arc estimates outward from a maximum point of $|Q_{n+1}|$, (56)--(57), gives $\frac{2(1-\varepsilon)}\pi\log n$, and letting $\varepsilon\to0$ gives (58).

Dependencies

Equation (28) of section 4 (printed p. 1038), used on p. 1047. The step on p. 1048 passes from $|S_n'(\theta)|>2/\lambda$ to $|S_n(\theta)|>2/(\lambda n)$ for some $\theta$, which is Bernstein's inequality for trigonometric sums; the paper does not name it. The matching upper bound for equally spaced points is attributed on p. 1041 to Grandjot (Jahresber. Deutsch. Math.-Verein. 34 (1925)), cited on p. 1026. The proof-scope page lists these interfaces.

Bears on

• Problem 1129: the algebraic consequence, as the paper states it, is the lower half of the asymptotic value of the minimal Lebesgue constant that the problem's nodes minimize; an asymptotic value does not describe the minimizing nodes, which is what the problem asks.
• Problem 1153: the algebraic consequence concerns the whole segment, the problem's case $a=-1$, $b=1$, with $n+1$ nodes where the problem has $n$. That page records the remarks of Erdős (1961) and Tao (2026) on whether Bernstein gave the algebraic case in full, and credits the instance to Erdős 1961.
• Problem 1132: the site's commentary there attributes a density statement to Bernstein. The Théorème bounds a maximum over a period at a point that may change with $n$; this page does not derive the site's statement from it.