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.
Statement
Let be a continuous function from to . Among all the polynomials of degree , the polynomial minimizes the uniform norm of the difference if and only if there are points such that where 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 and denominator has degree , the rational function , with and being relatively prime polynomials of degree and , minimizes the uniform norm of the difference if and only if there are points such that where is either -1 or +1.
Algorithms
Several minimax approximation algorithms are available, the most common being the Remez algorithm.