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Source. Theorem 2, 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 as in Theorem 1: a point group $x_i^{(n)}=\cos\vartheta_i^{(n)}$ and the counts $N_n(a,b)$.

Theorem 2 (p. 70). For a point group $x_i^{(n)}$ the following are equivalent.

• For every continuous $f(x)$ and every $c>0$ there is a sequence of polynomials $\psi_{n-1}(x)$ of degree $\le n-1$ with $\psi_{n-1}(x)\to f(x)$ uniformly in $(-1,+1)$ and $\psi_{n-1}(x_i^{(n)})=f(x_i^{(n)})$ for at least $n(1-c)$ values of $i$.
• For every $\varepsilon>0$,

$$\sum{}'N_n(a_k,b_k)=o(n)\qquad(11),$$

where $\sum'$ runs over an arbitrary set of disjoint "long" intervals (that is, $n(b_k-a_k)\to\infty$) satisfying

$$N(a_k,b_k)>\frac{n(b_k-a_k)}{\pi}(1+\varepsilon)\qquad(12);$$

and condition (10) of Theorem 1 is violated for at most $o(n)$ values of $i$.

The paper calls Theorem 2 a direct generalization of the theorem of S. Bernstein (its [1], 1932): for continuous $f$ on $[-1,1]$ and every $c>0$ there are polynomials $\psi_{n-1}$ of degree $\le n-1$ agreeing with $f$ at at least $n(1-c)$ roots of $T_n(x)$ and converging to $f$ uniformly in $(-1,+1)$. It summarizes Theorems 1 and 2 as requiring, roughly, that (9) and (10) be nearly always satisfied (p. 71).

Proof pointer

No proof in this paper. Erdős writes (p. 71) that Theorem 2 is not stated in his [8] (Annals of Math. 44 (1943), 330-337) but can be proved by its methods, and (p. 72) that it follows from Theorem 2'.

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

Bears on

No Erdős problem in the corpus.