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  "body": "\n## Statement\n\nLet $a \u003c b$, let $f \\colon [a, b] \\to \\R$ be continuous, and let $n \\ge 0$.\nLet $\\mathcal{P}_n$ be the set of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$, and let $\\norm{g} = \\max_{a \\le x \\le b} \\abs{g(x)}$.\nThen $p \\in \\mathcal{P}_n$ is a [[best approximation|7N4988HN3P]] of $f$ from $\\mathcal{P}_n$ if and only if there are $n + 2$ points $a \\le x_0 \u003c x_1 \u003c \\cdots \u003c x_{n+1} \\le b$ and a sign $\\sigma \\in \\set{-1, 1}$ with\n$$f(x_i) - p(x_i) = \\sigma (-1)^i \\norm{f - p} \\quad \\text{for } i = 0, 1, \\dots, n + 1.$$\nMoreover, the best approximation from $\\mathcal{P}_n$ exists and is unique.\n\n## Remarks\n\nThe idea of the proof: if the error $e = f - p$ alternates at $n + 2$ points with full size $\\norm{e}$, the [[de la Vallée Poussin lower bound|XNPVW5Z126]] gives $E_n(f) \\ge \\norm{e}$, so $p$ is best.\nConversely, suppose $e$ has fewer than $n + 2$ alternation points.\nThen one can split $[a, b]$ into at most $n + 1$ pieces on which the extreme values of $e$ have one sign.\nA polynomial $q$ of degree at most $n$ with the same sign pattern then makes $\\norm{f - p - t q} \u003c \\norm{e}$ for small $t \u003e 0$.\nUniqueness follows: if $p_1, p_2$ are best, then $(p_1 + p_2)/2$ is best, and at its $n + 2$ alternation points $p_1 = p_2$, so $p_1 - p_2$ has $n + 2$ zeros.\n\nExample: for $f(x) = x^2$ on $[0, 1]$ and $n = 1$, the line $p(x) = x - 1/8$ is the best approximation.\nThe error $x^2 - x + 1/8$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$, which is $n + 2 = 3$ alternating extreme values, and $\\norm{f - p} = 1/8$, so the [[error of best approximation|YDN0T4MEX0]] is $E_1(f) = 1/8$.\nAnother example: the monic polynomial of degree $n + 1$ of least norm on $[-1, 1]$ is $2^{-n} T_{n+1}$, where $T_{n+1}$ is the [[Chebyshev polynomial|M6EAR1JHPP]], because $T_{n+1}$ attains $\\pm 1$ alternately at $n + 2$ points.\n\n## Explanations\n\n### L1 human\n\nThe best polynomial approximation in the uniform sense is recognized by the shape of its error curve.\nThe error must reach its largest size $n + 2$ times, with alternating signs, like a wave of constant height.\nIf the error had fewer such peaks, you could tilt the polynomial a little and lower all the peaks at once.\n\n### L2 human\n\nThe theorem, due to Chebyshev, is the main characterization of minimax approximation by polynomials.\nIt is the basis of the Remez algorithm, which moves a set of $n + 2$ reference points until the error equioscillates.\nIt holds in the same form when $\\mathcal{P}_n$ is replaced by any Haar subspace of dimension $n + 1$ on $[a, b]$.\nIt also gives the extremal property of Chebyshev polynomials and many sharp constants in approximation theory.\n",
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