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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a [[continuous function|0X3WGX95FE]] on the [[interval|0TK8Y3J5RW]] $[a, b]$.\nLet $n \\ge 0$ be an integer and let $\\mathcal{P}_n$ be the set of real polynomials of [[degree|0DPF6WRZW6]] at most $n$.\nA *minimax approximation* of degree $n$ to $f$ is a polynomial $p^* \\in \\mathcal{P}_n$ such that\n$$\\max_{a \\le x \\le b} \\abs{f(x) - p^*(x)} = \\inf_{p \\in \\mathcal{P}_n} \\max_{a \\le x \\le b} \\abs{f(x) - p(x)}.$$\nThe common value is the *minimax error* $E_n(f)$.\n\n## Remarks\n\nExample: for $f(x) = x^2$ on $[-1, 1]$ and $n = 1$, the polynomial $p^*(x) = 1/2$ is the minimax approximation, with error $1/2$.\nCheck: $x^2 - 1/2$ takes the values $1/2, -1/2, 1/2$ at $x = -1, 0, 1$.\nIf a line $p$ had $\\abs{x^2 - p(x)} \u003c 1/2$ everywhere, then $p(-1) \u003e 1/2$, $p(0) \u003c 1/2$ and $p(1) \u003e 1/2$, which no linear function can do, since $p(0)$ is the mean of $p(-1)$ and $p(1)$.\n\nNon-example: the Taylor polynomial $1 + x$ of $e^x$ at $0$ is not the minimax approximation of degree $1$ on $[-1, 1]$.\nIts error $e^x - 1 - x$ is never negative, so lowering the line a little makes the largest error smaller.\n\nA minimax approximation exists and is unique for every continuous $f$.\nIt is characterised by equioscillation: the error $f - p^*$ attains its largest absolute value at $n + 2$ points with alternating signs.\nIn terms of norms, $p^*$ minimises $\\norm{f - p}_\\infty$ over $\\mathcal{P}_n$, where $\\norm{g}_\\infty = \\max_{[a, b]} \\abs{g}$.\n",
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