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]

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]

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

• Notes on how to prove Chebyshev’s equioscillation theorem at the Wayback Machine (archived July 2, 2011)
• The Chebyshev Equioscillation Theorem by Robert Mayans
• The de la Vallée-Poussin alternation theorem at the Encyclopedia of Mathematics
• Approximation theory by Remco Bloemen