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 ${\displaystyle a\leq x_{0}<x_{1}<\cdots <x_{n+1}\leq b}$ such that ${\displaystyle f(x_{i})-g(x_{i})=\sigma (-1)^{i}\|f-g\|_{\infty ))$ where ${\displaystyle \sigma }$ is either -1 or +1.^[1]

Algorithms

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