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  "body": "\n## Definition\n\nLet $k \\in L^1(\\R)$, a function in the [[$L^1$ space|SASP14MA1Z]] of $\\R$, and let $f$ be a given function on $[0, \\infty)$.\nA *Wiener–Hopf equation of the second kind* for an unknown function $\\varphi$ on $[0, \\infty)$ is\n$$\\varphi(x) - \\int_0^\\infty k(x - t)\\, \\varphi(t)\\, dt = f(x), \\quad x \\ge 0.$$\nA *Wiener–Hopf equation of the first kind* is\n$$\\int_0^\\infty k(x - t)\\, \\varphi(t)\\, dt = f(x), \\quad x \\ge 0.$$\nThe kernel $k(x - t)$ is a [[difference kernel|B5AB451TB7]], and both $x$ and $t$ range over the half-line $[0, \\infty)$.\n\n## Remarks\n\nThe unknown is usually sought in $L^p(0, \\infty)$ with $1 \\le p \\le \\infty$, or in a space of bounded or of polynomially growing functions.\nKernels such as $\\frac{1}{2} E_1(\\abs{x})$, with $E_1$ the [[exponential integral|EV6ZDQ0NDY]], are also used; this one is in $L^1(\\R)$.\n\nExample: $k(x) = \\frac{1}{2} e^{-\\abs{x}}$ and $f(x) = \\frac{1}{2} e^{-x}$.\nThe constant $\\varphi = 1$ is a solution.\nCheck: $\\int_0^\\infty \\frac{1}{2} e^{-\\abs{x - t}}\\, dt = \\frac{1}{2}(1 - e^{-x}) + \\frac{1}{2} = 1 - \\frac{1}{2} e^{-x}$, so $\\varphi - \\int_0^\\infty k(x - t) \\varphi(t)\\, dt = \\frac{1}{2} e^{-x}$.\nThe function $1 + x$ solves the homogeneous equation with the same kernel, so this equation has more than one solution among functions of linear growth.\n\nNon-example: $\\varphi(x) - \\int_{-\\infty}^{\\infty} k(x - t)\\, \\varphi(t)\\, dt = f(x)$ for all $x \\in \\R$ is a convolution equation on the whole line.\nIt is not a Wiener–Hopf equation, and the Fourier transform solves it directly when $1 - \\hat k$ has no zeros.\n\nThe integral operator $\\varphi \\mapsto \\int_0^\\infty k(x - t) \\varphi(t)\\, dt$ is the continuous analogue of a Toeplitz matrix, and it is called a Wiener–Hopf operator.\n",
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