Let be a continuous function from to . Among all the polynomials of degree , the polynomial minimizes the uniform norm of the difference if and only if there are points such that where is either -1 or +1.[1][2]
Variantsedit
The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree and denominator has degree , the rational function , with and being relatively prime polynomials of degree and , minimizes the uniform norm of the difference if and only if there are points such that where is either -1 or +1.[1]
equioscillation, theorem, mathematics, equioscillation, theorem, concerns, approximation, continuous, functions, using, polynomials, when, merit, function, maximum, difference, uniform, norm, discovery, attributed, chebyshev, contents, statement, variants, alg. In mathematics the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference uniform norm Its discovery is attributed to Chebyshev 1 Contents 1 Statement 2 Variants 3 Algorithms 4 References 5 External linksStatement editLet f displaystyle f nbsp be a continuous function from a b displaystyle a b nbsp to R displaystyle mathbb R nbsp Among all the polynomials of degree n displaystyle leq n nbsp the polynomial g displaystyle g nbsp minimizes the uniform norm of the difference f g displaystyle f g infty nbsp if and only if there are n 2 displaystyle n 2 nbsp points a x0 lt x1 lt lt xn 1 b displaystyle a leq x 0 lt x 1 lt cdots lt x n 1 leq b nbsp such that f xi g xi s 1 i f g displaystyle f x i g x i sigma 1 i f g infty nbsp where s displaystyle sigma nbsp is either 1 or 1 1 2 Variants editThe equioscillation theorem is also valid when polynomials are replaced by rational functions among all rational functions whose numerator has degree n displaystyle leq n nbsp and denominator has degree m displaystyle leq m nbsp the rational function g p q displaystyle g p q nbsp with p displaystyle p nbsp and q displaystyle q nbsp being relatively prime polynomials of degree n n displaystyle n nu nbsp and m m displaystyle m mu nbsp minimizes the uniform norm of the difference f g displaystyle f g infty nbsp if and only if there are m n 2 min m n displaystyle m n 2 min mu nu nbsp points a x0 lt x1 lt lt xn 1 b displaystyle a leq x 0 lt x 1 lt cdots lt x n 1 leq b nbsp such that f xi g xi s 1 i f g displaystyle f x i g x i sigma 1 i f g infty nbsp where s displaystyle sigma nbsp is either 1 or 1 1 Algorithms editSeveral minimax approximation algorithms are available the most common being the Remez algorithm References edit a b c Golomb Michael 1962 Lectures on Theory of Approximation Notes on how to prove Chebyshev s equioscillation theorem PDF Archived from the original PDF on 2 July 2011 Retrieved 2022 04 22 External links editNotes on how to prove Chebyshev s equioscillation theorem at the Wayback Machine archived July 2 2011 The Chebyshev Equioscillation Theorem by Robert Mayans The de la Vallee Poussin alternation theorem at the Encyclopedia of Mathematics Approximation theory by Remco Bloemen nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Equioscillation theorem amp oldid 1145933882, wikipedia, wiki, book, books, library,
