The Remez algorithm (Remez 1934), also called the Remez exchange algorithm, is an application of the Chebyshev alternation theorem that constructs the polynomial of best approximation to certain functions under a number of conditions. The Remez algorithm in effect goes a step beyond the minimax approximation algorithm to give a slightly finer solution to an approximation problem.
Parks and McClellan (1972) observed that a filter of a given length with minimal ripple would have a response with the same relationship to the ideal filter that a polynomial of degree ≤ n of best approximation has to a certain function, and so the Remez algorithm could be used to generate the coefficients.
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