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  "body": "\n## Summary\n\nA parametrix of a differential operator is an approximate inverse: an operator whose composites with the given operator differ from the identity by smoothing operators.\nFor operators on $\\R^n$ it is the same as an approximate [[fundamental solution|K2NK9A8M5F]], one whose error is a smooth function instead of zero.\nEvery [[elliptic operator|MT9P2X85V8]] has a parametrix, which can be built as a [[pseudo-differential operator|FF27N0BWTP]] from the inverse of the principal symbol.\nParametrices are the main tool for proving elliptic regularity and the Fredholm property of elliptic operators on compact manifolds.\n\n## Explanations\n\n### L1 human\n\nInverting a differential operator exactly is usually hard, and often impossible because the operator has a kernel.\nA parametrix gives up exactness but keeps what matters for smoothness: it inverts the operator up to an error that turns every input, however rough, into a smooth output.\nSuch errors are harmless for questions of regularity, and on a compact manifold they behave like finite-rank errors.\nFor the operator $-d^2/dx^2$ on the circle, dividing each Fourier coefficient $c_n$ with $n \\ne 0$ by $n^2$ is a parametrix, and its error only sees the average of the function.\n",
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