Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation:

Set $r_0 = f$.

for $k = 0,1,\ldots,$

Find the location of the global maximum of $|r_k|$, say $x_k\in[-1,1]$.

Let $p_k$ be the polynomial interpolant of $f$ at $x_0,\ldots,x_k$.

Set $r_{k+1} = f - p_k$.

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I am interested to know if $\|f - p_k\|_\infty\rightarrow0$ as $k\rightarrow \infty$. If $\|f - p_k\|_\infty\not\rightarrow0$ for all continuous $f$, does it when we assume $f$ is analytic on $[-1,1]$ and analytic continuable to some neighbourhood of $[-1,1]$?