analysis / Analysis, Polynomials
Let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$ on the 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 $\mathcal{L}^nf(x)$.
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An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
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Let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$ on the 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 $\mathcal{L}^nf(x)$.
An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
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