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  "body": "\n## Summary\n\nAn integral transform turns a function $f$ into a new function $Tf$ by integrating $f$ against a fixed function of two variables, the [[kernel|9M0NNNFJGM]].\nIn symbols, $(Tf)(u) = \\int f(t) K(t, u)\\, dt$.\nThe [[Laplace transform|0V49JEZW06]], the [[Fourier transform|DPVM34ADZV]] and the [[Mellin transform|N5PVGRJE1T]] are the main examples.\nA good transform turns a hard operation, such as differentiation or convolution, into an easy one, such as multiplication.\nIt can be undone by an [[inverse transform|GC96MQC0PP]].\n\n## Explanations\n\n### L1 human\n\nThink of the kernel as a family of test patterns, one for each value of $u$.\nThe number $(Tf)(u)$ measures how much $f$ looks like the pattern with label $u$.\nFor the Laplace transform the patterns are the decaying exponentials $e^{-st}$.\nFor example, the constant function $1$ on $[0, \\infty)$ has Laplace transform $\\int_0^\\infty e^{-st}\\, dt = 1/s$ for $s \u003e 0$.\nA problem about $f$ often becomes simpler when it is rewritten as a problem about $Tf$.\n",
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