{"schema":"vela.problem-packet.v0.1","problem":671,"statement":"Given $a_{i}^n\\in [-1,1]$ for all $1\\leq i\\leq n<\\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\\leq i'\\leq n$ with $i\\neq i'$. We similarly define\\[\\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i^n)p_i^n(x),\\]the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\\leq i\\leq n$ (that is, the sequence of Lagrange interpolation polynomials).Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists some $x\\in [-1,1]$ where\\[\\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty\\]and yet\\[\\mathcal{L}^nf(x) \\to f(x)?\\]Is there such a sequence such that\\[\\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty\\]for every $x\\in [-1,1]$ and yet for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists $x\\in [-1,1]$ with\\[\\mathcal{L}^nf(x) \\to f(x)?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}