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
Associated With
Includes
Source Wikipedia: Chebyshev polynomials
Source Wikipedia: Herman Müntz
Source Muntz-Szasz Theorem (Wikipedia)
Source Sergei Bernstein, Biography (MacTutor History of Mathematics)
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 Wikipedia: Herman Müntz
Includes: Herman Müntz, LeadQuote, Includes: Herman Müntz, Lead
In the following years he published on projective geometry, iterative methods, and approximation theory.
Wikipedia: Chebyshev polynomials
Includes: Chebyshev Polynomials, Lead paragraphQuote, Includes: Chebyshev Polynomials, Lead paragraph
In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation.
Sergei Bernstein, Biography (MacTutor History of Mathematics)
Includes: Sergei Bernstein, SummaryQuote, Includes: Sergei Bernstein, Summary
Sergei Bernstein was a Ukrainian mathematician who made important contributions to partial differential equations, differential geometry, probability theory and approximation theory.
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