Submit progress

on Erdős #1151 · analysis / polynomials

The statement

Given $a_1,\ldots,a_n\in [-1,1]$ let

\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i)\ell_i(x)

be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\leq i\leq n$ (that is, the Lagrange interpolation polynomial).

Let $a_i$ be the set of Chebyshev nodes. Prove that, for any closed $A\subseteq [-1,1]$ , there exists a continuous function $f$ such that $A$ is the set of limit points of $\mathcal{L}^nf(x)$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.