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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
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 Classification
Pure or AppliedBoth / Interdisciplinary 1 Connections
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Source Muntz-Szasz Theorem (Wikipedia)
Sources
1. Approximation Theory (Wikipedia)
WikipediaLead section
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.
Optimal polynomials
That such a polynomial is always optimal is asserted by the equioscillation theorem.
View the Source Stone-Weierstrass Theorem (Wikipedia)
WikipediaWeierstrass approximation theoremQuote, Weierstrass approximation theorem
The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.
Muntz-Szasz Theorem (Wikipedia)
Wikimedia FoundationIncludes: Muntz-Szasz Theorem, Lead sentenceView the Source Reader Challenges (0)
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