equioscillation theorem

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.{{Cite book |last=Golomb |first=Michael |url=https://books.google.com/books?id=DbnPAAAAMAAJ |title=Lectures on Theory of Approximation |year=1962}}

Statement

Let $f$ be a continuous function from $[a,b]$ to $\mathbb{R}$ . Among all the polynomials of degree $\le n$ , the polynomial $g$ minimizes the uniform norm of the difference $\| f - g \| _\infty$ if and only if there are $n+2$ points $a \le x_0 < x_1 < \cdots < x_{n+1} \le b$ such that $f(x_i) - g(x_i) = \sigma (-1)^i \| f - g \|_\infty$ where $\sigma$ is either -1 or +1.{{Cite web |title=Notes on how to prove Chebyshev's equioscillation theorem |url=http://www.math.uiowa.edu/~jeichhol/qual%20prep/Notes/cheb-equiosc-thm_2007.pdf |access-date=2022-04-22 |website= |archive-url=https://web.archive.org/web/20110702221651/http://www.math.uiowa.edu/~jeichhol/qual%20prep/Notes/cheb-equiosc-thm_2007.pdf |archive-date=2 July 2011 |url-status=dead}}

Variants

The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree $\le n$ and denominator has degree $\le m$ , the rational function $g = p/q$ , with $p$ and $q$ being relatively prime polynomials of degree $n-\nu$ and $m-\mu$ , minimizes the uniform norm of the difference $\| f - g \| _\infty$ if and only if there are $m + n + 2 - \min\{\mu,\nu\}$ points $a \le x_0 < x_1 < \cdots < x_{m+n+1 - \min\{\mu, \nu\}} \le b$ such that $f(x_i) - g(x_i) = \sigma (-1)^i \| f - g \|_\infty$ where $\sigma$ is either -1 or +1.

Algorithms

Several minimax approximation algorithms are available, the most common being the Remez algorithm.

References

External links

{{webarchive |url=https://web.archive.org/web/20110702221651/http://www.math.uiowa.edu/~jeichhol/qual%20prep/Notes/cheb-equiosc-thm_2007.pdf |date=July 2, 2011 |title=Notes on how to prove Chebyshev’s equioscillation theorem }}
[http://www.maa.org/publications/periodicals/loci/joma/the-chebyshev-equioscillation-theorem The Chebyshev Equioscillation Theorem by Robert Mayans]
[http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_theorem&oldid=44785 The de la Vallée-Poussin alternation theorem] at the Encyclopedia of Mathematics
[https://xn--2-umb.com/22/approximation/index.html Approximation theory by Remco Bloemen]

Category:Theorems about polynomials

Category:Numerical analysis

Category:Theorems in mathematical analysis