---

Statement

Setting as in the main Theorem (p. 232): nodes $x_{kn}=\cos t_{kn}$ with $0\le t_{1n}<\cdots<t_{nn}\le\pi$, the count $N_n(I)$ of angles in an interval $I\subseteq[0,\pi]$, the interpolation condition (2) $p(x_{kn})=f(x_{kn})$ for $k=1,\ldots,n$, the error $E_m(f)$ of best uniform approximation by polynomials of degree at most $m$, and the spacing condition (6), $\liminf_{n\to\infty}\min_{1\le i\le n-1}n(t_{i+1,n}-t_{i,n})>0$.

Theorem A (p. 241). Fix $d\ge1$. The following are equivalent for the array.

1. For every $f\in C[-1,1]$ and every $\varepsilon>0$ there is a sequence of polynomials $q_n\in\Pi_{[dn(1+\varepsilon)]}$ satisfying (2) and

$$\|f-q_n\|=O\bigl(E_{[dn(1+\varepsilon)]}(f)\bigr).$$

2. The array satisfies

$$\limsup_{n\to\infty}\frac{N_n(I_n)}{n|I_n|}\le\frac d\pi \quad\text{whenever}\quad\lim_{n\to\infty}n|I_n|=\infty,$$

and (6).

The case $d=1$ is the main Theorem.

Reading notes. The print places $d\ge1$ inside its opening quantifier ("For every $f(x)\in C[-1,1]$, $\varepsilon>0$, and $d\ge1$ there exists"), but the density bound $d/\pi$ in the second condition depends on $d$, so the equivalence is read for each fixed $d\ge1$; the print does not require $d$ to be an integer. The paper does not prove Theorem A: it introduces it with "Using the same arguments, we could have proved the following, slightly more general theorem" (p. 241) and gives no further argument.

Source. P. Erdős, A. Kroó and J. Szabados, On convergent interpolatory polynomials, Journal of Approximation Theory 58(2) (1989), 232--241, doi:10.1016/0021-9045(89)90022-1; Theorem A on p. 241. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. No proof is printed.

Proof pointer

None printed. The paper asserts that the arguments for the main Theorem (pp. 233--241: Lemmas 1 and 2 for sufficiency, the Bernstein-inequality argument for (6) and Lemma 3 for the density bound) carry over.

Dependencies

The proof of the main Theorem of the paper, which the authors say adapts.

Bears on

• Problem 1152: like the main Theorem, Theorem A concerns a fixed $\varepsilon>0$ (and a fixed $d\ge1$), not the regime $\varepsilon(n)\to0$ with failure of convergence almost everywhere that the problem asks about, and it does not answer the problem.