equioscillation theorem
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In mathematics, the equioscillation theorem concerns the approximation of continuous functions using
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s when the merit function is the maximum difference (
uniform norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
). Its discovery is attributed to Chebyshev.


Statement

Let f be a continuous function from ,b/math> to \mathbb. 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_ \le b such that f(x_i) - g(x_i) = \sigma (-1)^i \, f - g \, _\infty where \sigma is either -1 or +1.


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\ points a \le x_0 < x_1 < \cdots < x_ \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 algorithm A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that minimizes maximum error. For example, given a function f defined on the interval ,b/math> and ...
s are available, the most common being the
Remez algorithm The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are th ...
.


References


External links

*
The Chebyshev Equioscillation Theorem by Robert Mayans

The de la Vallée-Poussin alternation theorem
at the Encyclopedia of Mathematics

Theorems about polynomials Numerical analysis Theorems in analysis {{mathanalysis-stub