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  "body": "\n## Definition\n\nLet $J \\subseteq \\R$ be an interval, possibly unbounded.\nA *difference kernel* on $J$ is a [[kernel|GSMBXT7FPK]] $K \\colon J \\times J \\to \\C$ for which there is a function $k \\colon \\set{x - s : x, s \\in J} \\to \\C$ such that\n$$K(x, s) = k(x - s) \\quad \\text{for all } x, s \\in J.$$\n\n## Remarks\n\nExample: $K(x, s) = \\cos(x - s)$ with $k = \\cos$.\n\nNon-example: $K(x, s) = x s$ on $[0, 1]$.\nIf $x s = k(x - s)$, then putting $s = x$ gives $x^2 = k(0)$ for all $x \\in [0, 1]$, which is impossible.\n\nOn $J = \\R$, the integral operator is the [[convolution|2Z29JN3VES]] $\\int_{\\R} k(x - s) \\varphi(s) \\, ds = (k * \\varphi)(x)$.\nOn $J = [0, \\infty)$ with upper limit $x$, it is the one-sided convolution $\\int_0^x k(x - s) \\varphi(s) \\, ds$, which the Laplace transform turns into a product.\nEquations with a difference kernel are called *convolution equations*.\nThe equation $\\varphi(x) = f(x) + \\int_0^{\\infty} k(x - s) \\varphi(s) \\, ds$ for $x \\ge 0$ is the Wiener–Hopf equation.\n",
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