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  "body": "\n## Statement\n\nLet $\\mu$ be a probability measure on the unit circle with infinite support, and let $(\\alpha_n)_{n \\ge 0}$ be its [[Verblunsky coefficients|1RP3NVGNA6]].\nWrite the Lebesgue decomposition of $\\mu$ as $d\\mu = w(\\theta) \\, \\frac{d\\theta}{2\\pi} + d\\mu_s$, where $w \\ge 0$ is integrable and $\\mu_s$ is singular with respect to arc length.\nThen\n$$\\prod_{n=0}^{\\infty} \\big( 1 - \\abs{\\alpha_n}^2 \\big) = \\exp\\left( \\int_0^{2\\pi} \\log w(\\theta) \\, \\frac{d\\theta}{2\\pi} \\right),$$\nwhere both sides are $0$ when $\\int_0^{2\\pi} \\log w(\\theta) \\, d\\theta = -\\infty$.\nIn particular, $\\sum_{n} \\abs{\\alpha_n}^2 \u003c \\infty$ if and only if $\\int_0^{2\\pi} \\log w(\\theta) \\, d\\theta \u003e -\\infty$.\n\n## Remarks\n\nThe integral of $\\log w$ is well defined in $[-\\infty, 0]$, because Jensen's inequality gives $\\int \\log w \\, \\frac{d\\theta}{2\\pi} \\le \\log \\int w \\, \\frac{d\\theta}{2\\pi} \\le 0$.\nBy the norm formula for the [[monic orthogonal polynomials|3F9G3DRQ62]], the left side equals $\\lim_{n \\to \\infty} \\norm{\\Phi_n}^2$.\nAnd $\\norm{\\Phi_n}^2 = \\min \\set{ \\norm{p}^2 : p \\text{ monic of degree } n }$, so the theorem computes the best mean-square prediction error of $z^n$ from the past.\nA standard proof treats this minimum as an extremal problem in the Hardy space $H^2$ of the disk.\nAlong the way, the singular part $\\mu_s$ turns out not to affect the limit.\n\nEquivalently, in terms of the [[Toeplitz matrices|GKY2WSHE26]] $T_n = (c_{j-k})_{j,k=0}^{n}$ of the moments $c_k = \\int e^{-ik\\theta} \\, d\\mu$, one has $\\det T_n / \\det T_{n-1} = \\norm{\\Phi_n}^2$.\nSo the theorem also gives $\\lim_{n \\to \\infty} (\\det T_n)^{1/(n+1)} = \\exp \\int \\log w \\, \\frac{d\\theta}{2\\pi}$, Szegő's first limit theorem.\n\nExample: for $d\\mu = d\\theta/2\\pi$ one has $w = 1$ and all $\\alpha_n = 0$, and both sides equal $1$.\nIf $\\mu$ is singular, $w = 0$ almost everywhere, so $\\prod (1 - \\abs{\\alpha_n}^2) = 0$ and $\\sum \\abs{\\alpha_n}^2 = \\infty$.\n\n## Explanations\n\n### L1 human\n\nA measure on the circle is encoded by its Verblunsky coefficients, a sequence of points in the unit disk.\nSzegő's theorem links the size of these coefficients to how small the density of the measure can get.\nThe coefficients are square-summable exactly when the density is not too close to zero, in the sense that the integral of its logarithm is finite.\n\n### L2 human\n\nThe condition $\\int \\log w \u003e -\\infty$ is the Szegő condition.\nThe same condition governs the factorization $w = \\abs{D}^2$ with an outer function $D$ in the Hardy space.\nIt also governs prediction for stationary sequences: the future is not determined by the past exactly when it holds.\nThe theorem is a sum rule, relating the coefficients to the measure, and it is a starting point for finer results such as the [[Szegő limit theorems|WNJPRHZ680]] on Toeplitz determinants.\n",
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