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Source. Theorem 1, p. 311, 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

Setting (p. 309). For $n\ge1$ let $-1<x_1^{(n)}<x_2^{(n)}<\dots<x_n^{(n)}<1$ be the roots of the Chebyshev polynomial $T_n$, and for a function $f$ on $[-1,1]$ let $L_n(f(x))$ be the polynomial of degree at most $n-1$ that agrees with $f$ at these $n$ nodes. Its value at $x_0$ is $L_n(f(x_0))=\sum_k f(x_k^{(n)})\,l_k^{(n)}(x_0)$, where $l_k^{(n)}$ are the fundamental polynomials.

The point is $x_0=\cos\frac{p}{q}\pi$ with $p\equiv q\equiv1\pmod 2$, as fixed in the introduction (p. 309) and in Lemma 2 (p. 310). The introduction adds a coprimality condition printed as $(p_1,q)=1$; Lemma 2 and the proof do not use it. The proof of Lemma 2 places $x_0$ strictly between two consecutive nodes, so the argument treats points inside $(-1,1)$. When $p/q$ is an odd integer, $x_0=-1$; there the first bound of Lemma 2 fails, since the nearest node is at distance $1-\cos\frac{\pi}{2n}$, of order $1/n^2$, and the paper does not treat this point separately.

Theorem 1 (p. 311). For such $x_0$, "There exists a continuous function $f(x)$ such that $L_n(f(x_0))\to\infty$."

Remark (p. 313, unlabelled in the print). After the proof the paper states, without proof, that in the same way a continuous $f$ can be found for which $L_n(f(x_0))$ converges to any given value.

The nodes are symmetric about $0$, so the function $f(-x)$ gives divergence to infinity at $-x_0=\cos\frac{q-p}{q}\pi$, where the numerator $q-p$ is even, for every such $x_0$ inside $(-1,1)$. The paper does not state this case, and its Theorem 2 as printed contradicts it (see Theorem 2).

Proof pointer

Pp. 309--313. Lemma 1 (p. 309) bounds the distance between Chebyshev nodes of orders $m\ge n$ below by $1/m^3$; as printed it omits the needed hypothesis that the two nodes are distinct. At $x_0$ of the stated form inside $(-1,1)$, Lemma 2 (p. 310) gives constants with $\min_i|x_0-x_i^{(n)}|>c_1/n$ and $|T_n(x_0)|>c_2$. Lemma 3 (p. 310) bounds the sum of $|l_k^{(n)}(x_0)|$ over nodes not close to $x_0$ by a small power of $\log n$. Lemma 4 (p. 310) bounds single terms below, $|l_k^{(n)}(x_0)|>c_3/(j-k)$ for nodes between $0$ and $x_0$, and Lemma 5 (pp. 310--311) gives $\sum_{(2k-1,n)=1}|l_k^{(n)}(x_0)|>c_6\log n/\log\log n$, by a sieve count. The function is $f=\sum_{n\ge n_0}f_n/\sqrt{\log n}$, where $f_n$ is a narrow piecewise linear spike at each node $x_k^{(n)}$ with $(2k-1,n)=1$, of value the sign of $l_k^{(n)}(x_0)$. Lemma 1 makes the spikes of different orders disjoint enough for uniform convergence and for the later terms to vanish at the nodes of order $n$; the earlier terms are controlled by Lemma 3 and the $n$-th term by Lemma 5.

Read depth

Claims checked: the statement, the setting and the lemmas it rests on were read on the page images of the print, and the assembly of the proof was followed. The proof was not checked line by line.

Bears on

• Problem 1151: the problem page reads its Statement at a fixed $x_0=\cos(\pi p/q)$ with $p,q$ odd, the empty set meaning $\lvert\mathcal{L}^nf(x_0)\rvert\to\infty$. Theorem 1 gives a continuous $f$ with $L_n(f(x_0))\to\infty$ at every such point inside $(-1,1)$, the case of the empty set; its proof does not cover the point $x_0=-1$, where $p/q$ is an odd integer. The remark on p. 313, stated without proof, concerns convergence to a single given value; the paper treats no other closed set.