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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 $[-1,1]$?

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  • $\begingroup$ if $f$ is just continuous, having it be zero on some sub-interval should provide a counter example. $\endgroup$ – aginensky Nov 20 '13 at 14:39
  • $\begingroup$ Aginensky, I don't understand how this counter example is constructed. Can you say more? $\endgroup$ – alext87 Nov 20 '13 at 16:56
  • $\begingroup$ Sure, let $f=0$ for $-1\leq x \leq 0$ and let $f(x) = x^2$ for$ 0<x\leq 1$. That is a simple version of my proposed counter example. $\endgroup$ – aginensky Dec 5 '13 at 14:13

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