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  "body": "\n## Definition\n\nLet $\\partial\\mathbb{D} = \\set{z \\in \\C : \\abs{z} = 1}$ and let $\\mu$ be a Borel [[probability measure|0VGMEBVW1M]] on $\\partial\\mathbb{D}$ whose [[support|FQTXV6BQZG]] is an infinite set.\nGive the polynomials the inner product $\\ip{p}{q} = \\int_{\\partial\\mathbb{D}} p(z) \\overline{q(z)} \\, d\\mu(z)$ of the [[Lebesgue space|SASP14MA1Z]] $L^2(\\partial\\mathbb{D}, \\mu)$.\nFor $n \\ge 0$, the *monic orthogonal polynomial* $\\Phi_n$ of degree $n$ is the unique polynomial of the form $\\Phi_n(z) = z^n + \\text{(terms of lower degree)}$ such that\n$$\\ip{\\Phi_n}{z^k} = 0 \\quad \\text{for } k = 0, 1, \\dots, n - 1.$$\nThe *orthonormal polynomials* are $\\varphi_n = \\Phi_n / \\norm{\\Phi_n}$.\n\n## Remarks\n\nThe condition on the support makes $\\ip{p}{p} \u003e 0$ for every nonzero polynomial $p$, since a nonzero polynomial has finitely many zeros.\nThen $\\Phi_n$ is $z^n$ minus its orthogonal projection onto the span of $1, z, \\dots, z^{n-1}$, which shows existence and uniqueness.\nSuch measures are called *nontrivial*.\n\nExample: let $\\mu$ be the normalized arc length $d\\theta / 2\\pi$, with $z = e^{i\\theta}$.\nThen $\\ip{z^n}{z^k} = \\frac{1}{2\\pi} \\int_0^{2\\pi} e^{i(n - k)\\theta} \\, d\\theta = 0$ for $k \\ne n$, so $\\Phi_n(z) = z^n$ and $\\norm{\\Phi_n} = 1$.\n\nNon-example: let $\\mu = \\delta_1$, the point mass at $1$.\nIts support is finite, and the polynomial $z - 1$ has norm $0$.\nThe orthogonality conditions for $n = 1$ give $\\ip{z + c}{1} = 1 + c = 0$, so $\\Phi_1 = z - 1$, which is zero in $L^2(\\mu)$; the construction breaks down and does not give an orthonormal sequence.\n",
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