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.^[1]

Statement

Let ${\displaystyle f}$ be a continuous function from ${\displaystyle [a,b]}$ to ${\displaystyle \mathbb {R} }$. Among all the polynomials of degree ${\displaystyle \leq n}$, the polynomial ${\displaystyle g}$ minimizes the uniform norm of the difference ${\displaystyle \|f-g\|_{\infty }}$ if and only if there are ${\displaystyle n+2}$ points ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }}$ where ${\displaystyle \sigma }$ is either -1 or +1.^[1][2]

Variants

The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree ${\displaystyle \leq n}$ and denominator has degree ${\displaystyle \leq m}$, the rational function ${\displaystyle g=p/q}$, with ${\displaystyle p}$ and ${\displaystyle q}$ being relatively prime polynomials of degree ${\displaystyle n-\nu }$ and ${\displaystyle m-\mu }$, minimizes the uniform norm of the difference ${\displaystyle \|f-g\|_{\infty }}$ if and only if there are ${\displaystyle m+n+2-\min\{\mu ,\nu \}}$ points ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{m+n+1-\min\{\mu ,\nu \}}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty }}$ where ${\displaystyle \sigma }$ is either -1 or +1.^[1]

Algorithms

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