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  "body": "\n## Definition\n\nLet $\\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\\R$ such that:\n\n1. every moment $\\int_\\R \\abs{x}^k \\, d\\mu(x)$, $k = 0, 1, 2, \\dots$, is finite;\n2. the [[support|FQTXV6BQZG]] of $\\mu$ is an infinite set.\n\nA sequence $(p_n)_{n \\ge 0}$ of real [[polynomials|1G0B73WPX8]] is a sequence of *orthogonal polynomials* for $\\mu$ if $\\deg p_n = n$ for every $n$ and\n$$\\int_\\R p_m(x) \\, p_n(x) \\, d\\mu(x) = 0 \\quad \\text{for all } m \\ne n.$$\n\n## Remarks\n\nCondition 2 ensures that $\\int p^2 \\, d\\mu \u003e 0$ for every nonzero polynomial $p$, so the integral is an inner product on the polynomials.\nGram–Schmidt orthogonalization of $1, x, x^2, \\dots$ then gives such a sequence, and each $p_n$ is determined up to a nonzero constant factor.\nThe normalization $p_n(x) = x^n + \\cdots$ gives the *monic* orthogonal polynomials, and $\\int p_n^2 \\, d\\mu = 1$ gives the *orthonormal* ones.\nOften $d\\mu = w(x) \\, dx$ for a weight function $w \\ge 0$ on an interval.\n\nExample: $d\\mu = dx$ on $[-1, 1]$ and $p_0 = 1$, $p_1 = x$, $p_2 = \\frac12(3x^2 - 1)$, the first Legendre polynomials.\nThe integrals of $p_0 p_1 = x$ and of $p_1 p_2$ vanish because these are odd functions.\nAnd $\\int_{-1}^{1} \\frac12(3x^2 - 1) \\, dx = \\frac12 (2 - 2) = 0$.\n\nNon-example: the monomials $1, x, x^2$ for the same measure are not orthogonal, since $\\int_{-1}^{1} 1 \\cdot x^2 \\, dx = \\frac23 \\ne 0$.\n",
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