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  "body": "\n## Summary\n\nApproximation theory studies how well complicated functions can be replaced by simpler ones, such as polynomials, splines or rational functions.\nIt asks whether a [[best approximation|7N4988HN3P]] exists, whether it is unique, and how to recognize and compute it.\nIt also asks how fast the [[error of best approximation|YDN0T4MEX0]] decreases.\nIts results give the error bounds behind numerical methods for computing functions, integrals and solutions of equations.\n\n## Explanations\n\n### L1 human\n\nA computer can only add and multiply, so it evaluates $\\sin x$ or $e^x$ through a simpler formula, often a polynomial, that stays close to the true function.\nApproximation theory makes \"close\" precise, usually with a norm such as the largest gap between the two graphs.\nFor example, the [[Weierstrass approximation theorem|0JQE130DHW]] says that every continuous function on $[a, b]$ can be imitated by polynomials as closely as one likes.\nThe theory then asks which polynomial of a given degree is the best one and how small its error is.\n",
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