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Statement

Setting (printed p. 1027, section 2). For each $n$ let $b_i=\cos\big((i+\frac12)\frac\pi{n+1}\big)$, $i=0,\ldots,n$, be the zeros of $\cos\big((n+1)\arccos x\big)$, and take the nodes

$$a_i=b_i+\frac{\psi(b_i)}{n+1}\sqrt{1-b_i^2}\qquad(i=0,\ldots,n), \tag{3}$$

where $\psi$ is continuous on $[-1,1]$ (the paper suggests, for example, interpolating it linearly between consecutive $b_i$), $\psi(\pm1)=0$, and there are constants $A>0$ and $\delta>1$ with

$$|\psi(x)-\psi(y)|\le\frac{A}{|\log(x-y)|^{\delta}} \qquad(-1\le x\le1,\ -1\le y\le1). \tag{4}$$

The paper prints $\log(x-y)$ in (4); its later uses on pp. 1028--1029 apply it with a positive difference. In a parenthesis on printed pp. 1029--1030 it allows $\psi$ to depend on $n$: the $\psi$ in the limit formulas (11) and (12) is the limit of these functions, and it satisfies (4) when the constant $A$ there does not depend on $n$.

Conclusion (printed p. 1036, end of section 3). For these nodes the Lebesgue function $F$ of equation (1) has all its $n+2$ maxima, one on each of $(-1,a_0),(a_0,a_1),\ldots,(a_n,1)$, asymptotically equal to $\frac2\pi\log n$ as $n\to\infty$. Displays (26) and (26 bis) give the interior maxima and the values $F(\pm1)$ respectively.

On the way the paper states asymptotic formulas for the nodal polynomial and its derivative at the nodes, uniformly on the segment: with $p(x)=\exp\big(\frac1\pi\int_0^\pi \frac{\psi(\cos\theta)\sin\theta-\psi(\cos\varphi)\sin\varphi} {\cos\theta-\cos\varphi},d\varphi\big)$, $x=\cos\theta$, from (13) on p. 1030,

$$A_{n+1}(z)\sim\frac{p(z)}{2^n}\cos(n+1)\arccos \Big[z-\frac{\psi(z)\sqrt{1-z^2}}{n+1}\Big] \quad\text{(17 bis, p. 1032)}, \qquad \lim\frac{2^nA_{n+1}'(a_h)}{n+1}\sqrt{1-a_h^2}=(-1)^hp(a_h) \quad\text{(19, p. 1033)}.$$

The first expression in (17 bis) uses a function $\psi_1$ close to $\psi$ and prints the denominator $n-1$ where (17) has $n+1$. A footnote on p. 1032 gives the analogous asymptotic outside the segment under weaker conditions on $\psi$.

The case $\psi\equiv0$ is the Chebyshev system $a_i=b_i$, the nodal polynomial $C\cos(n+1)\arccos x$ named on pp. 1025--1026. Bernstein introduces the class on p. 1027 as polynomials for which all $n+2$ maxima are asymptotic to $\frac2\pi\log n$; with the Théorème this gives the asymptotic (2), $M\sim\frac2\pi\log n$.

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; section 2 on printed pp. 1027--1033 and section 3 on pp. 1033--1036 (PDF pp. 3--12). The copy read is identified on the source card.

Read depth. Claims checked: the hypotheses (3)--(4), the remark on pp. 1029--1030, displays (17 bis), (19), (26), (26 bis) and the closing conclusion were read on the page images. The proofs were read for their structure and not checked; this page reconstructs none of them.

Proof pointer

Pages 1028--1036. Writing the nodal polynomial at the shifted point as a product over the Chebyshev zeros, (5)--(7), the paper bounds each factor's deviation from $1$ by (8)--(9) using (4), so that the logarithm of the product converges to the singular integral (11) and the product to $p$, (13). The change of variable (15)--(16) gives (17) and (17 bis), and a product computation gives the derivative asymptotic (18)--(19). Section 3 inserts these into $F$, (20)--(21), locates the nodes and the maximum points through (22)--(24), and evaluates the resulting sums as integrals, (25)--(26) and (26 bis), each $\sim\frac2\pi\log n$. Passing from (20) to (21) assumes, as printed on p. 1034, that $F(x)$ grows indefinitely with $n$.

Dependencies

Bernstein's 1930 memoir Polynômes orthogonaux relatifs à un segment fini, Journ. de Math., chapter II, section 9, cited on p. 1030 for the convergence of the singular integral under (4); the uniformly convergent sine series of $\psi(\cos\theta)$ and its conjugate series, used on pp. 1030--1031. The proof-scope page lists these interfaces.

Bears on

• Problem 1129: the class shows, as the paper states it, that the minimal Lebesgue constant is at most $(\frac2\pi+o(1))\log n$, and that many node systems, not only the Chebyshev one, have all interval maxima asymptotically equal. Asymptotic equality of the maxima is weaker than the exact equality in Bernstein's conjecture, and nothing here describes the exact minimizers.
• Problem 1153: for these node systems the maximum of $F$ over the whole segment is $(\frac2\pi+o(1))\log n$ by the paper's conclusion, so its maximum over any fixed $[a,b]$ is at most that, and the coefficient $\frac2\pi$ in that problem cannot be raised. The paper has $n+1$ nodes where the problem has $n$, which does not change the asymptotic.