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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $C[a, b]$ be the space of continuous functions $[a, b] \\to \\R$ with the supremum norm.\nLet $L_n \\colon C[a, b] \\to C[a, b]$, $n \\ge 1$, be [[positive linear operators|AQDXWH4812]]: each $L_n$ is linear, and $L_n f \\ge 0$ whenever $f \\ge 0$.\nLet $e_0(x) = 1$, $e_1(x) = x$ and $e_2(x) = x^2$.\nIf $L_n e_i \\to e_i$ [[uniformly|08FPABDQ4C]] on $[a, b]$ for $i = 0, 1, 2$, then $L_n f \\to f$ uniformly on $[a, b]$ for every $f \\in C[a, b]$.\n\n## Remarks\n\nThe idea of the proof: fix $f$ and $\\varepsilon \u003e 0$.\nBy [[uniform continuity|0Q4DR37DVE]] there is $\\delta \u003e 0$ with $\\abs{f(t) - f(x)} \\le \\varepsilon + 2 \\norm{f}_\\infty (t - x)^2 / \\delta^2$ for all $t, x \\in [a, b]$.\nApply the positive operator $L_n$ in the variable $t$ and evaluate at $x$.\nThis bounds $\\abs{L_n f(x) - f(x)}$ by $\\varepsilon$ plus terms in $L_n e_0 - e_0$, $L_n e_1 - e_1$ and $L_n e_2 - e_2$, which tend to $0$ uniformly.\n\nExample: the Bernstein operators $B_n$ on $[0, 1]$ satisfy $B_n e_0 = e_0$, $B_n e_1 = e_1$ and $B_n e_2 = e_2 + x(1 - x)/n$.\nSo Korovkin's theorem gives $B_n f \\to f$ uniformly for every continuous $f$.\n\nPositivity is needed.\nIn the periodic setting the test functions are $1, \\cos x, \\sin x$.\nThe partial sums of Fourier series reproduce these test functions for $n \\ge 1$, but they are not positive operators.\nAnd there are continuous periodic functions whose Fourier series do not converge uniformly.\n\n## Explanations\n\n### L1 human\n\nMany approximation schemes are positive linear operators: they average values of $f$ with nonnegative weights.\nKorovkin's theorem says that to check such a scheme works for every continuous function, it is enough to check three functions: $1$, $x$ and $x^2$.\nThis turns a hard question into a short computation.\n\n### L2 human\n\nThe theorem is the standard tool to prove convergence of the Bernstein, Szász–Mirakyan, Baskakov and Kantorovich operators and of Fejér means.\nThe periodic version uses the test functions $1, \\cos x, \\sin x$, and it proves Fejér's theorem on Cesàro means of Fourier series.\nQuantitative forms bound $\\norm{L_n f - f}_\\infty$ by the modulus of continuity of $f$ and the errors on the test functions.\nThe theorem also shows a rigidity: a single positive linear operator on $C[a, b]$ that fixes $1$, $x$ and $x^2$ must be the identity.\n",
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