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  "body": "\n## Statement\n\nLet $a \u003c b$.\nFor each $n \\ge 0$, let $x_{n,0} \u003c x_{n,1} \u003c \\cdots \u003c x_{n,n}$ be $n + 1$ points of $[a, b]$.\nFor a [[continuous function|0X3WGX95FE]] $f$ on $[a, b]$, let $p_n(f)$ be the [[Lagrange interpolation|237KR6FRCK]] polynomial of degree at most $n$ that agrees with $f$ at $x_{n,0}, \\dots, x_{n,n}$.\nThen there is a continuous function $f$ on $[a, b]$ such that $p_n(f)$ does not [[converge uniformly|08VZ57PT26]] to $f$ on $[a, b]$.\n\n## Remarks\n\nProof idea: the norm of the interpolation map $f \\mapsto p_n(f)$ on $C[a, b]$ is the [[Lebesgue constant|7V7KWNK9HN]] $\\Lambda_n$ of the $n$-th row of nodes.\nFor every choice of nodes, $\\Lambda_n \\ge c \\log n$ for some constant $c \u003e 0$, so $\\Lambda_n \\to \\infty$.\nBy the uniform boundedness principle, $\\sup_n \\norm{p_n(f)}_\\infty = \\infty$ for some continuous $f$, and for this $f$ the sequence $p_n(f)$ cannot converge uniformly.\n\nExample: for Chebyshev nodes $\\Lambda_n$ grows only like $\\frac{2}{\\pi} \\log n$, which is slow but still unbounded.\nSo even interpolation at Chebyshev nodes fails for some continuous function, although it converges for every Lipschitz function.\n\nThe theorem concerns one fixed array of nodes for all functions.\nFor each single continuous $f$ there is an array of nodes for which $p_n(f) \\to f$ uniformly, for example nodes where $f$ meets its best approximations.\n\n## Explanations\n\n### L1 human\n\nNo fixed recipe for choosing interpolation points works for every continuous function.\nWhatever points are chosen, some continuous function makes the interpolating polynomials fail to converge to it.\nThis contrasts with the Weierstrass approximation theorem, which says that good polynomial approximations always exist.\n\n### L2 human\n\nFaber's theorem separates existence of good approximations from their construction by interpolation.\nIt is a typical application of the uniform boundedness principle with norms that grow like a logarithm, as for divergent Fourier series of continuous functions.\nIn practice it is mild: for Chebyshev nodes, Lebesgue's lemma gives convergence as soon as $\\omega(f, 1/n) \\log n \\to 0$.\n",
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