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Statement

Setting (p. 232). The nodes form a triangular array

$$X_n:\ -1\le x_{nn}<x_{n-1,n}<\cdots<x_{1n}\le1\qquad(n=1,2,\ldots),$$

written $x_{kn}=\cos t_{kn}$ with $0\le t_{1n}<t_{2n}<\cdots<t_{nn}\le\pi$. For an interval $I\subseteq[0,\pi]$, $N_n(I)$ is the number of the $t_{kn}$ lying in $I$, and $|I|$ is its length. $\Pi_m$ is the set of algebraic polynomials of degree at most $m$, $\|\cdot\|$ is the maximum norm on $[-1,1]$, and $E_m(f)$ is the error of best uniform approximation of $f$ by $\Pi_m$. Interpolation at the nodes is condition (2): $p_n(x_{kn})=f(x_{kn})$ for $k=1,\ldots,n$ and $n=1,2,\ldots$.

Theorem (pp. 232--233; the paper's main result, printed without a number). The following are equivalent for the array $X_n$.

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

$$\|f-p_n\|=O\bigl(E_{[n(1+\varepsilon)]}(f)\bigr),$$

the paper's (4), where the $O$ refers to $n\to\infty$ and its constant depends only on $\varepsilon$. 2. The array satisfies both

$$\limsup_{n\to\infty}\frac{N_n(I_n)}{n|I_n|}\le\frac1\pi \quad\text{whenever}\quad\lim_{n\to\infty}n|I_n|=\infty,$$

the paper's (5), for intervals $I_n\subseteq[0,\pi]$, and

$$\liminf_{n\to\infty}\ \min_{1\le i\le n-1}n(t_{i+1,n}-t_{i,n})>0,$$

the paper's (6).

So (5) bounds the density of the angles $t_{kn}$ on every interval long compared with $1/n$ by the density $1/\pi$ of the Chebyshev angles, and (6) keeps consecutive angles at least a constant multiple of $1/n$ apart for all large $n$.

Reading notes. The print defines $E_m(f)$ as the best approximation "by polynomials of degree at most $n$" [sic] (p. 233); the degree meant is $m$. The paper records (p. 233) that the theorem, with (4) replaced by plain uniform convergence $\|f-p_n\|\to0$ (its (3)), was stated without proof as Theorem 4 of Erdős's 1943 paper (Ann. of Math. (2) 44 (1943), 330--337), and that this paper supplies the proof. The proof of necessity uses only weaker consequences of statement 1: the necessity of (6) uses boundedness of $\|p_n\|$, and the necessity of (5) uses (4) for a family of functions with uniformly bounded best-approximation errors.

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; the statement on pp. 232--233, the proof on pp. 233--241. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read for its structure, not checked step by step. Nothing here is independently reviewed.

Proof pointer

Sufficiency (pp. 233--238). Lemma 1 (pp. 233--236) embeds the given angles, under (5) and (6), into a system of $m=[n(1+\varepsilon)]$ angles $\eta_k=\frac{2k-1+d_k}{m}\frac\pi2$ that are separated by $c/n$ with an absolute constant $c$ and whose perturbations have partial sums $\bigl|\sum_{k\le s}d_k\bigr|\le A(\varepsilon)$. Lemma 2 (pp. 236--238) shows that the Lagrange fundamental polynomials of this enlarged system are uniformly bounded, by comparison with the Chebyshev nodes and Fejér's bound $\sqrt2$ for theirs. The interpolant (p. 238) applies Lemma 1 with $\varepsilon/3$, corrects a best approximation by Lagrange interpolation on the enlarged system, and damps each correction with squared sums of adjacent Lagrange fundamental polynomials on the $s=[n\varepsilon/3]$ Chebyshev nodes, using the Erdős--Turán lower bound for such sums (Lemma IV of On interpolation III); the degree stays below $n(1+\varepsilon)$ and the error is $O(E_{[n(1+\varepsilon)]}(f))$.

Necessity of (6) (p. 239). If gaps $n(t_{i_n+1,n}-t_{i_n,n})$ tend to $0$, a continuous $f$ rising by $\sqrt{\varepsilon_n}$ across gaps of length at most $\varepsilon_n/n$ forces, by Bernstein's inequality, $|p_n|\ge 1/\sqrt{\varepsilon_n}\to\infty$, contradicting (4).

Necessity of (5) (pp. 239--241). Lemma 3 (pp. 239--240): if trigonometric polynomials of order at most $r_n\uparrow\infty$ are bounded by $M$ and $r_n|I_n|\to\infty$, the number $Q(I_n)$ of their alternating $\pm1$ oscillations on $I_n$ satisfies $\limsup Q(I_n)/(r_n|I_n|)\le1/\pi$. Applying it to interpolants of the functions $f_n(x)=F_n(\arccos x)$, where $F_n$ is piecewise linear in the angle and takes the values $(-1)^k$ at the angles $t_{kn}$, whose best-approximation errors are bounded, gives (5) with $[(1+\varepsilon)n]$ in place of $n$ in the denominator, and letting $\varepsilon\to0$ gives (5).

Dependencies

Lemmas 1, 2 and 3 of the paper; Fejér's bounds for the Lagrange fundamental polynomials on the Chebyshev nodes; the Erdős--Turán Lemma IV (Ann. of Math. (2) 41 (1940), 510--553, the paper's [2]); Bernstein's inequality.

Bears on

• Problem 1152: the problem takes an arbitrary array and $\varepsilon=\varepsilon(n)\to0$ and asks whether some continuous $f$ makes every sequence of interpolants of degree below $(1+\varepsilon(n))n$ fail to converge to $f$ at almost every point of $[-1,1]$. The theorem treats a fixed $\varepsilon>0$ instead: for every array satisfying (5) and (6), every continuous $f$ has interpolants of degree at most $[n(1+\varepsilon)]$ converging uniformly to $f$. It says nothing about the regime $\varepsilon(n)\to0$ the problem asks about and does not answer the problem.