# Equioscillation theorem

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 $f$ be a continuous function from $[a,b]$ to $\mathbf{R}$. 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 $\sigma = \pm 1$.

## Algorithms

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