Equioscillation theorem

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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.


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.


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