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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to [0, \\infty)$ be a $2\\pi$-periodic function, [[integrable|X7GTF3D98F]] on $[-\\pi, \\pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\\hat f(k) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(\\theta) e^{-ik\\theta} \\, d\\theta$.\nFor $n \\ge 1$ let $T_n(f) = (\\hat f(j - k))_{j,k=0}^{n-1}$ be the $n \\times n$ [[Toeplitz matrix|GKY2WSHE26]] of $f$, and let $D_n(f) = \\det T_n(f)$ be its [[determinant|00KDZRW4C0]].\n\n1. (First Szegő theorem.) If $\\log f$ is integrable, then\n$$\\lim_{n \\to \\infty} D_n(f)^{1/n} = G(f), \\qquad G(f) = \\exp\\left( \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\log f(\\theta) \\, d\\theta \\right).$$\n2. (Strong Szegő theorem.) Assume $f = e^{g}$ with $g$ real-valued and continuous, with Fourier coefficients satisfying $\\sum_{k \\in \\Z} \\abs{\\hat g(k)} \u003c \\infty$ and $\\sum_{k \\in \\Z} \\abs{k} \\abs{\\hat g(k)}^2 \u003c \\infty$.\nThen\n$$\\lim_{n \\to \\infty} \\frac{D_n(f)}{G(f)^n} = \\exp\\left( \\sum_{k=1}^{\\infty} k \\, \\hat g(k) \\hat g(-k) \\right) = \\exp\\left( \\sum_{k=1}^{\\infty} k \\abs{\\hat g(k)}^2 \\right).$$\n\n## Remarks\n\nProof idea for the first theorem: the ratio $D_{n+1}(f) / D_n(f)$ is the minimum of $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{p(e^{i\\theta})}^2 f(\\theta) \\, d\\theta$ over polynomials $p(z) = 1 + a_1 z + \\cdots + a_n z^n$.\nThese minima decrease to $G(f)$ by [[Szegő's theorem|SZ8ARMDEMY]] on prediction, and then the geometric means $D_n^{1/n}$ have the same limit.\nThe strong theorem gives the next term of the asymptotics, $D_n(f) \\sim E(f) \\, G(f)^n$ with the constant $E(f)$ above.\n\nExample: let $f(\\theta) = \\abs{1 - a e^{i\\theta}}^2$ with $\\abs{a} \u003c 1$.\nThen $T_n(f)$ is tridiagonal with diagonal $1 + \\abs{a}^2$, and $D_n(f) = \\frac{1 - \\abs{a}^{2(n+1)}}{1 - \\abs{a}^2}$.\nHere $g = \\log f$ has $\\hat g(0) = 0$ and $\\hat g(k) = -a^k / k$ for $k \\ge 1$, so $G(f) = 1$ and the strong theorem predicts the limit $\\exp\\left(\\sum_{k \\ge 1} \\abs{a}^{2k}/k\\right) = \\frac{1}{1 - \\abs{a}^2}$, as the formula confirms.\n\nSzegő proved the first theorem in 1915 and the strong theorem in 1952, motivated by the computation of the spontaneous magnetization of the two-dimensional Ising model.\nThe strong theorem was later proved under weaker hypotheses, essentially only the condition $\\sum \\abs{k} \\abs{\\hat g(k)}^2 \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nA Toeplitz matrix has constant entries along each diagonal, and these entries are the Fourier coefficients of a function on the circle.\nThe Szegő theorems say how the determinants of larger and larger such matrices grow.\nTo first order, they grow like $G^n$, where $G$ is the geometric mean of the function.\nThe strong theorem gives the precise constant factor in front.\n\n### L2 human\n\nThe theorems are the analogue for Toeplitz matrices of the fact that the determinant of a diagonal matrix is the product of its entries.\nThe first theorem is closely tied to [[orthogonal polynomials on the unit circle|3F9G3DRQ62]] and to prediction theory of stationary sequences.\nThe strong theorem has applications in statistical mechanics and in random matrix theory, where such determinants compute averages over the unitary group.\n",
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