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  "body": "\n## Definition\n\nLet $(m_n)_{n \\ge 0}$ be a sequence of real numbers.\nFor $n \\ge 0$ let $\\Delta_n = \\det (m_{i+j})_{i,j=0}^{n}$ be the [[determinant|00KDZRW4C0]] of its $(n+1) \\times (n+1)$ [[Hankel matrix|PQ1D8FQB3E]], and put $\\Delta_{-1} = 1$.\nAssume that $\\Delta_n \u003e 0$ for all $n \\ge 0$.\nThe *moment functional* of $(m_n)$ is the [[linear form|07YQ1C78G0]] $L$ on the real polynomials $\\R[x]$ with $L(x^n) = m_n$ for all $n \\ge 0$.\nThe *monic orthogonal polynomials* of $(m_n)$ are the polynomials $P_0 = 1$ and, for $n \\ge 1$,\n$$P_n(x) = \\frac{1}{\\Delta_{n-1}} \\det \\begin{pmatrix} m_0 \u0026 m_1 \u0026 \\cdots \u0026 m_n \\\\ m_1 \u0026 m_2 \u0026 \\cdots \u0026 m_{n+1} \\\\ \\vdots \u0026 \\vdots \u0026 \u0026 \\vdots \\\\ m_{n-1} \u0026 m_n \u0026 \\cdots \u0026 m_{2n-1} \\\\ 1 \u0026 x \u0026 \\cdots \u0026 x^n \\end{pmatrix}.$$\nEach $P_n$ is monic of degree $n$, and $L(P_m P_n) = 0$ for $m \\ne n$.\n\n## Remarks\n\nOrthogonality holds because, for $k \u003c n$, applying $L$ to $x^k P_n$ replaces the last row by $(m_k, \\ldots, m_{n+k})$, which equals row $k$, so the determinant vanishes.\nThe leading coefficient is $1$ since the cofactor of $x^n$ is $\\Delta_{n-1}$.\nAlso $L(P_n^2) = \\Delta_n / \\Delta_{n-1} \u003e 0$.\nLet $(m_n)$ be the moment sequence of a positive Borel measure $\\mu$ on $\\R$ with infinite support and all moments finite.\nThen $\\Delta_n \u003e 0$ for all $n$, and the $P_n$ are the monic [[orthogonal polynomials|A18HPFPC4D]] of $\\mu$ in $L^2(\\mu)$.\n\nExample: for Lebesgue measure on $[-1, 1]$ one has $m_0 = 2$, $m_1 = 0$, $m_2 = \\tfrac23$ and $\\Delta_0 = 2$, $\\Delta_1 = \\tfrac43$.\nThe formula gives $P_1(x) = \\tfrac12 \\det \\begin{pmatrix} 2 \u0026 0 \\\\ 1 \u0026 x \\end{pmatrix} = x$ and\n$P_2(x) = \\tfrac34 \\left( \\tfrac43 x^2 - \\tfrac49 \\right) = x^2 - \\tfrac13$, the monic Legendre polynomials.\n\nNon-example: the moments of the Dirac measure at $0$ are $m_0 = 1$ and $m_n = 0$ for $n \\ge 1$.\nThen $\\Delta_1 = \\det \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix} = 0$, so the condition $\\Delta_n \u003e 0$ fails.\nIndeed $L(x^2) = 0$, and no monic polynomial of degree $1$ is orthogonal to $1$ with nonzero norm.\n\n## Explanations\n\n### L1 human\n\nAll the information needed to build orthogonal polynomials sits in the moments.\nThe determinant formula turns a list of moments directly into the polynomials, one degree at a time.\n",
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