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  "body": "\n## Statement\n\nLet $M$ be a smooth real submanifold of $\\C^n$ that is generic, that is, $T_q M + J\\, T_q M = T_q \\C^n$ for every $q \\in M$, where $J$ is multiplication by $i$.\nLet $p \\in M$.\nThen there is a compact neighbourhood $K$ of $p$ in $M$ with the following property.\nLet $f$ be a continuous [[CR function|JFKYYK2CEB]] on $M$.\nThen there is a sequence $(P_k)$ of complex [[polynomials|07GNX8GCYP]] in $z_1, \\dots, z_n$ such that $P_k \\to f$ uniformly on $K$.\n\n## Remarks\n\nThe theorem is due to M. S. Baouendi and F. Trèves, who proved it for locally integrable structures; the statement above is its form for embedded [[CR manifolds|ZC78JKZX9S]].\nThe proof convolves $f$ with the Gaussian kernel $e^{-\\tau \\sum_j (z_j - \\zeta_j)^2}$ over a suitable totally real piece of $M$.\nThe result is an entire function of $z$.\nThe CR equations and Stokes' theorem show that it tends to $f$ on $K$ as $\\tau \\to \\infty$.\nTruncating the power series of these entire functions then gives polynomials.\nThe neighbourhood $K$ depends on $M$ and $p$ but not on $f$.\nFor example, let $M = \\C^{n-1} \\times \\R \\subset \\C^n$.\nIts CR functions are the functions holomorphic in $(z_1, \\dots, z_{n-1})$ for each fixed real $z_n$.\nThe theorem says that they are local uniform limits of holomorphic polynomials in all $n$ variables.\nThe result is local: a CR function on all of $M$ need not be a uniform limit of polynomials on large compact subsets of $M$.\n\n## Explanations\n\n### L1 human\n\nCR functions on a submanifold of complex space are defined by a system of differential equations, and they can be quite rough.\nThe theorem says that near each point they can nevertheless be approximated as closely as one wants by ordinary polynomials in the complex coordinates.\nSo every CR function is locally a limit of restrictions of holomorphic functions on the whole space.\n\n### L2 human\n\nThis approximation theorem is a basic tool in the local theory of CR functions.\nBy the maximum principle, polynomials that converge uniformly on $K$ also converge uniformly on the polynomially convex hull $\\widehat{K}$ of $K$.\nSo every continuous CR function on $M$ extends holomorphically to the interior of $\\widehat{K}$, which gives a way to prove extension results such as the [[Lewy extension theorem|V2YCGHD719]].\nThe theorem also shows that CR functions in this sense are exactly the local uniform limits of restrictions of holomorphic functions.\n",
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