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  "body": "\n## Summary\n\nThe Fourier transform method solves a [[convolution integral equation|EY75YD988Z]] on the whole line or on $\\R^n$ by passing to the frequency side.\nThe [[convolution theorem|QY8ETVDA1Z]] turns the convolution with the kernel into multiplication by the transform of the kernel.\nSo the integral equation becomes an algebraic equation for the transform of the unknown.\nOne solves it by division and returns with the [[inverse Fourier transform|JB8QMT8DCH]].\nThe method also shows when the equation has exactly one solution in $L^2$ for every right side in $L^2$: the symbol $1 - \\lambda \\hat k(\\xi)$ must not vanish.\n\n## Explanations\n\n### L1 human\n\nA convolution mixes the values of a function along the whole line, which makes the equation hard to solve point by point.\nThe [[Fourier transform|DPVM34ADZV]] splits a function into pure waves $e^{i x \\xi}$, and a convolution acts on each wave by a simple factor.\nSo in the wave picture the equation $\\varphi - k * \\varphi = f$ becomes $(1 - \\hat k(\\xi)) \\hat\\varphi(\\xi) = \\hat f(\\xi)$, one equation for each frequency $\\xi$.\nFor example, with $k(s) = \\frac{1}{2} e^{-\\abs{s}}$ one has $\\hat k(\\xi) = \\frac{1}{1 + \\xi^2}$, and dividing by $1 - \\hat k(\\xi) = \\frac{\\xi^2}{1 + \\xi^2}$ is a problem only at $\\xi = 0$.\n",
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