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  "body": "\n## Statement\n\nLet $k \\in L^1(\\R)$ and write $\\hat g(\\xi) = \\int_{\\R} g(x)\\, e^{i\\xi x}\\, dx$ for the [[Fourier transform|DPVM34ADZV]].\nAssume that $1 - \\hat k(\\xi) \\ne 0$ for all $\\xi \\in \\R$.\nSince $\\hat k(\\xi) \\to 0$ as $\\abs{\\xi} \\to \\infty$, the curve $\\xi \\mapsto 1 - \\hat k(\\xi)$, $\\xi \\in [-\\infty, \\infty]$, is closed; let $\\nu$ be its [[winding number|0X7SBS8E0F]] about $0$.\nLet $r(\\xi) = (\\xi - i)/(\\xi + i)$.\n\n1. There are $k_+, k_- \\in L^1(\\R)$, with $k_+ = 0$ on $(-\\infty, 0)$ and $k_- = 0$ on $(0, \\infty)$, and\n$$1 - \\hat k(\\xi) = \\bigl(1 - \\hat k_-(\\xi)\\bigr)\\, r(\\xi)^{\\nu}\\, \\bigl(1 - \\hat k_+(\\xi)\\bigr), \\quad \\xi \\in \\R.$$\nHere $1 - \\hat k_+$ has no zeros in the closed upper half-plane and $1 - \\hat k_-$ has no zeros in the closed lower half-plane.\n2. Let $1 \\le p \u003c \\infty$ and $W \\varphi(x) = \\varphi(x) - \\int_0^\\infty k(x - t) \\varphi(t)\\, dt$ on $L^p(0, \\infty)$. Then $W$ is a [[Fredholm operator|8HWGKWD6ZM]] of [[index|2V9Q4KHY1T]] $-\\nu$. If $\\nu = 0$ it is invertible. If $\\nu \u003e 0$ it is injective and its range has codimension $\\nu$. If $\\nu \u003c 0$ it is surjective and its kernel has dimension $-\\nu$.\n\n## Remarks\n\nThe functions $1 - \\hat k_\\pm$ extend [[holomorphically|0NAHXR1FRH]] to the corresponding half-planes, because $k_\\pm$ vanish on a half-line.\nWhen $\\nu = 0$, a continuous branch of $\\log\\bigl(1 - \\hat k\\bigr)$ on $\\R$ tends to $0$ at infinity; splitting it into an upper and a lower part and exponentiating gives the two factors.\nThe difficult point is that the split pieces again come from $L^1$ functions, which uses [[Wiener's lemma|NC768TWZ3D]] on absolutely convergent Fourier integrals.\nFor $\\nu \\ne 0$ one first divides by $r^\\nu$, whose winding number is $\\nu$.\n\nExample: $k(x) = 2 e^{-x}$ for $x \u003e 0$ and $k = 0$ otherwise gives $1 - \\hat k(\\xi) = 1 - \\frac{2}{1 - i\\xi} = \\frac{\\xi - i}{\\xi + i} = r(\\xi)$.\nSo $\\nu = 1$, $k_\\pm = 0$, and $W$ is injective with range of codimension $1$.\n\nThe hypothesis $1 - \\hat k \\ne 0$ on $\\R$ is needed: for $k(x) = \\frac{1}{2} e^{-\\abs{x}}$ the symbol $\\frac{\\xi^2}{1 + \\xi^2}$ vanishes at $0$, and then $W$ is not Fredholm on $L^p(0, \\infty)$.\n\n## Explanations\n\n### L1 human\n\nA Wiener–Hopf equation is a convolution equation cut off at $0$.\nIts Fourier symbol can be written as a product of a part that \"looks only forward\" and a part that \"looks only backward\".\nA simple extra factor counts how many times the symbol winds around $0$.\nThis winding number decides everything: if it is $0$ the equation has exactly one solution for each right side.\n\n### L2 human\n\nThe theorem is the continuous analogue of the theory of Toeplitz operators, where the symbol lives on the unit circle instead of the real line.\nIt turns the Wiener–Hopf method into a complete solvability theory for kernels in $L^1(\\R)$.\nThe sign convention of the Fourier transform matters.\nWith the kernel $e^{-i\\xi x}$ the half-planes swap roles and the winding number changes sign, so the index becomes $+\\nu$.\nFor systems of equations the symbol is a matrix function, and a single winding number is replaced by several partial indices, which is the matrix Wiener–Hopf factorization.\n",
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