Branches of Mathematics
Approximation Theory
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Approximation theory is concerned with how functions can best be approximated by simpler functions, typically polynomials, and with quantifying the error such an approximation introduces.
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Central QuestionGiven a continuous function on a closed interval, how closely can it be approximated by a polynomial of a given degree, and which polynomial of that degree achieves the smallest possible error. 1 Key DebateWhether an optimal approximating polynomial can be certified rather than merely constructed. The equioscillation theorem answers this for polynomial approximation on an interval, but the underlying fact, that continuous functions can be approximated as closely as desired by polynomials at all, proved by Karl Weierstrass in 1885, took decades of further work after Weierstrass's own proof before the sharpest, provably optimal constructions followed. 1 Cross-Tradition Connections
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