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  "body": "\n## Summary\n\nAn integral equation of the first kind is a [[linear integral equation|AYZCKG34QK]] in which the unknown function appears only under the integral: $\\int K(x, s) \\varphi(s) \\, ds = f(x)$.\nSolving it means inverting an integral operator.\nSuch equations are typically ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\\varphi$.\nInverse problems, such as recovering a signal from blurred data, lead to equations of the first kind.\n\n## Explanations\n\n### L1 human\n\nOne only sees a smoothed version $f$ of the unknown $\\varphi$, and one wants $\\varphi$ back.\nSmoothing destroys fine detail, so going back amplifies noise.\nFor example, $\\int_0^x \\varphi(s) \\, ds = f(x)$ asks for a function whose integral is $f$.\nIts continuous solution is $\\varphi = f'$, which exists only when $f$ is continuously differentiable with $f(0) = 0$.\n",
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