analysis / Analysis, Interpolation
For triangular arrays of nodes $a_i^n\in[-1,1]$ let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$, with fundamental polynomials $p_i^n$. Is there a choice of nodes such that for every continuous $f$ there is some $x$ where $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ and yet $\mathcal{L}^nf(x) \to f(x)$? Is there a choice with $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ for every $x$, yet for every continuous $f$ some $x$ has $\mathcal{L}^nf(x)\to f(x)$? Both questions are claimed resolved in the affirmative.
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Append-only history
Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open
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For triangular arrays of nodes $a_i^n\in[-1,1]$ let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$, with fundamental polynomials $p_i^n$. Is there a choice of nodes such that for every continuous $f$ there is some $x$ where $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ and yet $\mathcal{L}^nf(x) \to f(x)$? Is there a choice with $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ for every $x$, yet for every continuous $f$ some $x$ has $\mathcal{L}^nf(x)\to f(x)$? Both questions are claimed resolved in the affirmative.
Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open
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