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  "body": "\n## Definition\n\nLet $I$ and $J$ be intervals of $\\R$, and let $T$ be an [[integral transform|YCK709QB06]] given by\n$$(Tf)(u) = \\int_I f(t)\\, K(t, u)\\, dt \\quad (u \\in J)$$\nfor a measurable function $K \\colon I \\times J \\to \\C$.\nThe function $K$ is the *kernel* (or *nucleus*) of $T$.\n\n## Remarks\n\nExample: the Laplace transform $F(s) = \\int_0^\\infty f(t) e^{-st}\\, dt$ has kernel $K(t, s) = e^{-st}$ on $[0, \\infty) \\times (0, \\infty)$; this is read off by comparing with the formula above.\nNon-example: the function $f$ itself is not a kernel of $T$; the kernel does not depend on the input $f$, only on the transform.\nSuppose the transform is defined on all bounded functions with compact support.\nThen the kernel is determined by the transform up to changes on a set of measure zero in $I \\times J$.\nA kernel is *symmetric* when $I = J$ and $K(t, u) = K(u, t)$.\n",
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