Equioscillation theorem

Statement

Let f be a continuous function from [ a , b ] to R . Among all the polynomials of degree ≤ n , the polynomial g minimizes the uniform norm of the difference | | f − g | | ∞ if and only if there are n + 2 points a ≤ x 0 < x 1 < ⋯ < x n + 1 ≤ b such that f ( x i ) − g ( x i ) = σ ( − 1 ) i | | f − g | | ∞ where σ = ± 1 .

Algorithms

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