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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] of functions, and let $A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\cdots$ be nonempty subsets of $X$, the approximating classes.\n*Approximation theory* is the study of the following questions for $f \\in X$.\n\n1. Existence: does $f$ have a [[best approximation|7N4988HN3P]] from $A_n$?\n2. Uniqueness and characterization: is it unique, and which conditions single it out?\n3. Rates: how fast does the [[error of best approximation|YDN0T4MEX0]] $E(f, A_n)$ tend to $0$ as $n \\to \\infty$, and how does this rate depend on $f$?\n4. Construction: which explicit methods give good [[approximations|VNWH2KADN7]] of $f$ from $A_n$?\n\n## Remarks\n\nExample: $X = C[a, b]$ with the supremum norm and $A_n$ the polynomials of degree at most $n$.\nHere a best approximation exists and is unique, it is characterized by the equioscillation theorem, and $E(f, A_n) \\to 0$ for every $f$ by the Weierstrass approximation theorem.\n\nNon-example: a single numerical estimate, such as $\\abs{e^x - (1 + x)} \\le 1$ on $[0, 1]$, is an approximation result but does not by itself pose these questions.\nThe theory studies whole families $A_n$ and all $f$ in a class.\n\nOther common choices are trigonometric polynomials, splines with $n$ knots, rational functions of degree $n$, and $L^p$ norms instead of the supremum norm.\n",
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