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Source. Theorem 3, pp. 71-72 (notation p. 71), of P. Erdős, "Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems," Mathematica (Cluj) 10 (33) (1968), 65-73; see the source card.

Statement

Notation (p. 71). For nodes $-1\le x_1^{(m)}<\cdots<x_m^{(m)}\le1$ write $x_i^{(m)}=\cos\vartheta_i^{(m)}$ (the print writes $\cos x_i^{(m)}=\vartheta_i^{(m)}$ [sic], introduced as repeating the convention of p. 70). Let $\alpha\le\vartheta_i^{(m)}<\vartheta_{i+1}^{(m)}<\cdots<\vartheta_j^{(m)}\le\beta$ be the $\vartheta^{(m)}$ in $(\alpha,\beta)$, so that $N_m(\alpha,\beta)=j-i+1$. For $\eta>0$ select a subsequence greedily: $\vartheta_{i_1}^{(m)}=\vartheta_i^{(m)}$, and $\vartheta_{i_{r+1}}^{(m)}$ is the least $\vartheta_s^{(m)}\ge\vartheta_{i_r}^{(m)}+\eta/m$. This gives $\vartheta_{i_1}^{(m)}<\cdots<\vartheta_{i_l}^{(m)}$ with $\vartheta_{i_l}^{(m)}>\vartheta_j^{(m)}-\eta/m$; put $N_m^{(\eta)}(\alpha,\beta)=l$.

Theorem 3 (pp. 71-72). Let $-1\le x_1^{(m)}<\cdots<x_m^{(m)}\le1$, $m=1,2,\ldots$, and let $P_n(x)$ be a polynomial of degree $n$ with

$$|P_n(x_i^{(m)})|\le1,\quad i=1,\ldots,m,\quad m>n(1+c)\qquad(14).$$

The condition that (14) implies, for every $c>0$,

$$\max_{-1\le x\le1}|P_n(x)|<A(c)\qquad(15)$$

holds if and only if there is an $\eta>0$, independent of $m$, such that for every $\alpha_m<\beta_m$ with $m(\beta_m-\alpha_m)\to\infty$,

$$N_m^{(\eta)}(\alpha_m,\beta_m)\ge(1+o(1))\frac m\pi(\beta-\alpha)\qquad(16).$$

The right side of (16) is printed with $\beta-\alpha$, without the index $m$. The paper glosses (16): every interval large compared to $1/m$ contains asymptotically at least as many points $\vartheta_i^{(m)}$, no two of them too close, as the roots of $\cos mx$.

The paper presents Theorem 3 as a comprehensive generalization of a result of S. Bernstein (its [2], 1931): if $m>n(1+c)$, the $x_i$, $1\le i\le m$, are the roots of $T_m(x)$, and $|P_n(x_i)|\le1$ for $i=1,\ldots,m$, then $\max_{-1\le x\le1}|P_n(x)|<A=A(c)$; and of Zygmund's result (its [23]) for the roots of the Legendre polynomial (p. 71).

Proof pointer

No proof in this paper; the theorem is from its [9], P. Erdős, On the boundedness and unboundedness of polynomials, Journal d'Analyse 18.

Read depth. The statement and notation were read clause by clause on the printed pages.

Bears on

• Problem 1133: background. By the paper's remark under Theorem 4, the hypothesis $m>n(1+c)$ cannot be weakened to $m>n(1+o(1))$.