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Source. Theorem 4, p. 72, 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

Theorem 4 (p. 72). For every $A$, however large, there is an $\varepsilon>0$ such that if $n>n_0(A,\varepsilon)$ and $m=[(1+\varepsilon)n]$, then for every $-1\le x_1<\cdots<x_m\le1$ there is a polynomial $P_n(x)$ of degree $n$ with

$$|P_n(x_i)|\le1,\quad i=1,\ldots,m,\qquad\text{and}\qquad \max_{-1\le x\le1}|P_n(x)|>A.$$

The paper says the theorem sharpens a result of Faber (its [12], 1914), which is the case $m=n+1$, and shows that in Theorem 3 the hypothesis $m>n(1+c)$ can never be weakened to $m>n(1+o(1))$.

Proof pointer

None. Erdős writes that he states Theorem 4 without proof in his [9], P. Erdős, On the boundedness and unboundedness of polynomials, Journal d'Analyse 18; this paper gives no proof either.

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

Bears on

• Problem 1133: a weaker statement, announced without proof. The paper says the conjecture on p. 72, which is the problem's assertion, would contain Theorem 4. Theorem 4 does not imply the problem's assertion.