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Source. The closing remark, p. 315, of P. Erdős, On divergence properties of the Lagrange interpolation parabolas, Ann. of Math. (2) 42 (1941), 309--315, doi:10.2307/1968999; the edition read is named on the source card.

Statement

The setting is that of Theorem 1: $L_m(f(x))$ is the Lagrange interpolation polynomial of $f$ at the roots of the Chebyshev polynomial $T_m$.

Closing remark (p. 315, unlabelled in the print). For every $x_0$ in $(-1,+1)$ there is a continuous $f$ with

$$\lim_{n\to\infty}\frac{1}{n}\sum_{m\le n}L_m(f(x_0))=\infty.$$

The print names the point $x$ in the quantifier and $x_0$ in the formula.

Proof pointer

None given: the paper says only that the proof is very similar to that of Theorem 1.

Read depth

Claims checked: the statement was read on the page image of the print. There is no proof to check.

Bears on

None directly. Problem 1151 asks for the limit points of the interpolation polynomials themselves, and the remark concerns their arithmetic means.