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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be a measurable set, and let $U$ be a set.\nLet $K \\colon U \\times \\Omega \\to \\C$ be a function, the [[kernel|9M0NNNFJGM]].\nLet $f \\colon \\Omega \\to \\C$.\nIts *multidimensional transform* with kernel $K$ is\n$$(Tf)(u) = \\int_\\Omega K(u, x)\\, f(x)\\, dx,$$\na [[multiple integral|FE1ZZJ8QW2]], defined at those $u \\in U$ where the integral converges absolutely.\nFor functions on $\\Z^n$ the integral is replaced by a sum over $\\Z^n$.\n\n## Remarks\n\nThe transform is *separable* when $\\Omega = \\Omega_1 \\times \\cdots \\times \\Omega_n$, $U = U_1 \\times \\cdots \\times U_n$ and $K(u, x) = K_1(u_1, x_1) \\cdots K_n(u_n, x_n)$.\nThen, by Fubini's theorem, $T$ is the composition of the one-variable transforms with kernels $K_j$ applied in each variable.\n\nExample: $\\Omega = U = \\R^n$ and $K(\\xi, x) = e^{-i \\xi \\cdot x}$ give the $n$-dimensional [[Fourier transform|DPVM34ADZV]], a separable transform since $e^{-i \\xi \\cdot x} = \\prod_j e^{-i \\xi_j x_j}$.\n\nNon-example: the [[Radon transform|VJNZJ352PC]] sends a function on $\\R^n$ to its integrals over hyperplanes.\nIt is not of the form above with a function $K$, since each value is an integral over a set of measure zero.\n",
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