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Source. Theorem 1, p. 70, 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. 70). A point group is a triangular array $x_i^{(n)}$, $i=1,\ldots,n$, $n=1,2,\ldots$, of nodes in $[-1,1]$. Put $\cos\vartheta_i^{(n)}=x_i^{(n)}$, and for $0\le a<b\le\pi$ let $N_n(a,b)$ be the number of the $\vartheta_i^{(n)}$ in $(a,b)$.

Theorem 1 (p. 70). Let $x_i^{(n)}$ be a point group. The following are equivalent.

• For every continuous function $f(x)$ and every $c>0$ there is a sequence of polynomials $\varphi_m(x)$ of degree $m<n(1+c)$ with $\varphi_m(x_i^{(n)})=f(x_i^{(n)})$ for $i=1,\ldots,n$ and $\varphi_m(x)\to f(x)$ uniformly in $(-1,+1)$.
• Whenever $n(b_n-a_n)\to\infty$ with $0\le a_n<b_n\le\pi$,

$$\limsup_{n\to\infty}\frac{N_n(a_n,b_n)}{n(b_n-a_n)}\le\frac1\pi \qquad(9),$$

and

$$\liminf_{n\to\infty}n\bigl(\vartheta_{i+1}^{(n)}-\vartheta_i^{(n)}\bigr)>0, \qquad i=1,\ldots,n\qquad(10).$$

Both conditions are as printed; (10) is printed with $i$ running to $n$, although $\vartheta_{n+1}^{(n)}$ is not defined. The sentence after the theorem explains condition (9), which the print calls "Condition (1)" [sic]: the number of $\vartheta_i^{(n)}$ in a long interval $(a_n,b_n)$ cannot be much larger than the number of roots of $\cos nx=0$ there.

The paper says Theorem 1 is related to, but does not generalize, a theorem of S. Bernstein (its [1]) on interpolation at $n(1-c)$ of the roots of $T_n(x)$; see Theorem 2.

Proof pointer

No proof in this paper. The theorem is from the paper's [8], P. Erdős, On some convergence properties of the interpolation polynomials, Annals of Math. 44 (1943), 330-337. The paper says (p. 71) that Erdős's results with Turán (cited there as [9], the paper's reference to Erdős's own Journal d'Analyse paper) imply (9) and (10) under the assumption (13), $|l_k(x)|<c_5$ for $-1\le x\le1$, $k=1,\ldots,n$, $n=1,2,\ldots$; that [8] proves Theorem 1 under (13) in a very simple way; and that the proof of Theorems 1 and 2 in full generality is rather complicated. It also says (p. 72) that Theorems 1 and 2 follow from Theorem 1' and Theorem 2'.

Read depth. The statement was read clause by clause on the printed page.

Bears on

No Erdős problem in the corpus.