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  "body": "\n## Definition\n\nLet $I$ and $J$ be intervals of $\\R$, with $-\\infty \\le t_1 \u003c t_2 \\le \\infty$ the endpoints of $I$.\nLet $K \\colon I \\times J \\to \\C$ be a measurable function.\nLet $f \\colon I \\to \\C$ be a [[function|0XHDTE5YMZ]].\nThe *integral transform* of $f$ with kernel $K$ is the function $Tf$ given by\n$$(Tf)(u) = \\int_{t_1}^{t_2} f(t)\\, K(t, u)\\, dt$$\nat each $u \\in J$ for which the integral converges.\nThe integral is a [[Lebesgue integral|XVEWR93EMV]] or an improper Riemann integral, depending on the setting.\n\n## Remarks\n\nExample: take $I = [0, \\infty)$, $J = (0, \\infty)$ and $K(t, u) = e^{-ut}$.\nThis is the Laplace transform; for $f = 1$ the integral $\\int_0^\\infty e^{-ut}\\, dt = 1/u$ converges for every $u \\in J$.\nNon-example: the map $f \\mapsto f'$ is linear but is not an integral transform with a function kernel.\nThe value $f'(u)$ depends only on the values of $f$ near $u$, and no integrable kernel produces this for all smooth $f$.\nThe map $f \\mapsto Tf$ is linear wherever both sides converge.\nThe same definition works with $I$ and $J$ replaced by measure spaces, for example subsets of $\\R^n$.\n",
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