Law of cotangents: Difference between revisions
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The '''equioscillation theorem''' concerns the [[Approximation theory|approximation]] of [[continuous function]]s using [[polynomial]]s when the merit function is the maximum difference ([[uniform norm]]). Its discovery is attributed to [[Pafnuty Chebyshev|Chebyshev]]. | |||
== Statement == | |||
Let <math>f</math> be a continuous function from <math>[a,b]</math> to <math>\mathbf{R}</math>. Among all the polynomials of degree <math>\le n</math>, the polynomial <math>g</math> minimizes the uniform norm of the difference <math> || f - g || _\infty </math> if and only if there are <math>n+2</math> points <math>a \le x_0 < x_1 < \cdots < x_{n+1} \le b</math> such that <math>f(x_i) - g(x_i) = \sigma (-1)^i || f - g || _\infty</math> where <math>\sigma = \pm 1</math>. | |||
== Algorithms == | |||
Several [[minimax approximation algorithm]]s are available, the most common being the [[Remez algorithm]]. | |||
== References == | |||
* [http://www.math.uiowa.edu/~jeichhol/qual%20prep/Notes/cheb-equiosc-thm_2007.pdf Notes on Notes on how to prove Chebyshev’s equioscillation theorem] | |||
* [http://mathdl.maa.org/images/upload_library/4/vol6/Mayans/Contents.html Another The Chebyshev Equioscillation Theorem by Robert Mayans] | |||
[[Category:Polynomials]] | |||
[[Category:Numerical analysis]] | |||
[[Category:Theorems in analysis]] | |||
{{mathanalysis-stub}} | |||
Revision as of 19:37, 30 October 2013
Template:Multiple issues 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.
Statement
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 .
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
Algorithms
Several minimax approximation algorithms are available, the most common being the Remez algorithm.