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  "body": "\n## Summary\n\nA Fredholm equation of the first kind is an integral equation $\\int_a^b K(x, y) \\varphi(y) \\, dy = f(x)$ with fixed limits $a$ and $b$.\nThe unknown function $\\varphi$ appears only under the integral sign.\nIt is the [[Fredholm integral equation|526TCVMSHP]] form of an [[integral equation of the first kind|ZDS77Q0F06]].\nSuch equations model the recovery of a cause from a smoothed measurement, as in deblurring and tomography.\nThey are usually ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\\varphi$.\n\n## Explanations\n\n### L1 human\n\nThink of $f$ as a blurred photograph and of $\\varphi$ as the sharp picture that produced it.\nThe kernel $K(x, y)$ says how much the point $y$ of the sharp picture contributes to the point $x$ of the blurred one.\nSolving the equation means undoing the blur, and blurring destroys fine detail, so this is hard.\nFor example, with $K(x, y) = 1$ on $[0, 1]$, the equation reads $\\int_0^1 \\varphi(y) \\, dy = f(x)$.\nIt has a solution only when $f$ is constant, and then it has infinitely many.\n",
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  "name": "Fredholm equation of the first kind",
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