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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\C^n$ be a bounded domain with $C^2$ boundary $\\partial\\Omega$, and let $\\sigma$ be the [[surface measure|065HE5V98E]] on $\\partial\\Omega$.\nLet $H^2(\\partial\\Omega)$ be the closure in $L^2(\\partial\\Omega, \\sigma)$ of the restrictions to $\\partial\\Omega$ of the functions [[holomorphic|507C34W0ZT]] on a neighbourhood of $\\overline{\\Omega}$.\nEach $f \\in H^2(\\partial\\Omega)$ is the boundary value of a unique holomorphic function on $\\Omega$, also written $f$.\nFor each $z \\in \\Omega$, the map $f \\mapsto f(z)$ is a bounded linear functional on the Hilbert space $H^2(\\partial\\Omega)$.\nThe *Szegő kernel* of $\\Omega$ is the function $S \\colon \\Omega \\times \\partial\\Omega \\to \\C$ such that, for each $z \\in \\Omega$, the function $\\zeta \\mapsto \\overline{S(z, \\zeta)}$ lies in $H^2(\\partial\\Omega)$ and\n$$f(z) = \\int_{\\partial\\Omega} S(z, \\zeta) \\, f(\\zeta) \\, d\\sigma(\\zeta) \\quad \\text{for all } f \\in H^2(\\partial\\Omega).$$\n\n## Remarks\n\nBy the Riesz representation theorem the function $\\overline{S(z, \\cdot)}$ exists and is unique.\nSo $S$ is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] $H^2(\\partial\\Omega)$, in the sense of [[reproducing kernel Hilbert spaces|4CK2GSRAGD]].\nThe kernel extends to $\\Omega \\times \\Omega$, holomorphic in the first variable and antiholomorphic in the second, with $S(z, w) = \\overline{S(w, z)}$.\n\nExample: for the unit disc $\\mathbb{D} \\subset \\C$ with arc length, $S(z, \\zeta) = \\dfrac{1}{2\\pi(1 - z\\bar\\zeta)}$.\nCheck: for $\\abs{\\zeta} = 1$ one has $\\bar\\zeta = 1/\\zeta$, so $\\dfrac{1}{1 - z\\bar\\zeta} = \\dfrac{\\zeta}{\\zeta - z}$, and $d\\sigma = \\abs{d\\zeta} = \\dfrac{d\\zeta}{i\\zeta}$.\nThe formula becomes $f(z) = \\dfrac{1}{2\\pi i}\\displaystyle\\oint_{\\abs{\\zeta} = 1} \\dfrac{f(\\zeta)}{\\zeta - z}\\, d\\zeta$, which is Cauchy's integral formula.\nFor the unit ball of $\\C^n$ the kernel is $S(z, \\zeta) = \\dfrac{(n-1)!}{2\\pi^n} (1 - \\ip{z}{\\zeta})^{-n}$, where $\\ip{z}{\\zeta} = \\sum_j z_j \\bar\\zeta_j$; at $z = 0$ it is $1$ divided by the area $2\\pi^n/(n-1)!$ of the sphere.\n\nNon-example: on the unit disc, the Poisson kernel $P(z, \\zeta) = \\dfrac{1 - \\abs{z}^2}{2\\pi\\abs{\\zeta - z}^2}$ also satisfies $f(z) = \\int_{\\partial\\mathbb{D}} P(z, \\zeta) f(\\zeta) \\, d\\sigma(\\zeta)$ for all $f \\in H^2(\\partial\\mathbb{D})$.\nIt is not the Szegő kernel, because for $z \\ne 0$ the function $\\zeta \\mapsto P(z, \\zeta)$ has nonzero Fourier coefficients of negative index, so it does not lie in $H^2(\\partial\\mathbb{D})$.\n\nThe [[Bergman kernel|8RWDHW0CXC]] is defined in the same way, with $L^2(\\Omega)$ in place of $L^2(\\partial\\Omega, \\sigma)$.\n",
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