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  "body": "\n## Summary\n\nA convolution integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel depends only on the difference $x - t$ of its two variables.\nThe integral in it is then a [[convolution|2Z29JN3VES]] of the kernel with the unknown function.\nBecause integral transforms turn convolutions into products, such equations can often be solved explicitly with the [[Fourier transform|DPVM34ADZV]] or the [[Laplace transform|0V49JEZW06]].\nThey model systems that are invariant under translation, such as linear filters, renewal processes and homogeneous media.\n\n## Explanations\n\n### L1 human\n\nThink of a signal that passes through a blurring device that treats every position in the same way.\nThe output at a point is a weighted average of the input around that point, with weights that depend only on the distance.\nAsking for the input that produced a given output is a convolution integral equation.\nFor example, $\\varphi(x) - \\frac{1}{2} \\int_{-\\infty}^{\\infty} e^{-\\abs{x - t}} \\varphi(t) \\, dt = f(x)$ asks for $\\varphi$ when $f$ is known.\nSince the weight $e^{-\\abs{x - t}}$ depends only on $x - t$, the equation is of convolution type.\n",
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