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  "body": "\n## Summary\n\nA difference kernel is a [[kernel|GSMBXT7FPK]] of the form $K(x, s) = k(x - s)$, which depends only on the difference of its two variables.\nIts integral operator is a [[convolution|2Z29JN3VES]] with $k$, so it commutes with translations.\nIntegral equations with difference kernels are called convolution equations.\nTransforms that turn convolutions into products, such as the [[Laplace transform|0V49JEZW06]] and the Fourier transform, reduce them to algebraic equations.\n\n## Explanations\n\n### L1 human\n\nThe influence of the point $s$ on the point $x$ depends only on how far apart they are, and in which direction.\nShifting both points by the same amount does not change the kernel.\nFor example, $K(x, s) = e^{-\\abs{x - s}}$ is a difference kernel with $k(t) = e^{-\\abs{t}}$.\nThe Volterra equation $\\varphi(x) = f(x) + \\int_0^x (x - s) \\varphi(s) \\, ds$ has the difference kernel with $k(t) = t$.\n",
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