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Universal Interpolation

A. Vogt

If $P_n$ is the polynomial of degree at most $n-1$ which interpolates a function $f:[0,1]\to\mathbb{R}$ at the nodes $0\leq x^n_1<x_2^n<\ldots<x_n^n\leq1\,\,(n\in\mathbb{N})$, it is well-known that, even if $f$ is a continuous function, the sequence $(P_n)_{n\in\mathbb{N}}$ does not necessarily converge to $f$. Indeed, for $p\in[1,\infty)$, there exists an infinitely often differentiable function $f$ and a ``nice'' system of nodes such that to every measurable function $g$, there exists a subsequence of $(P_n)_{n\in\mathbb{N}}$ that converges in $L_p$ to $g$.