Analytical Approximations to Approximations in the Chebyshev Sense
S. Darlington · Bell System Technical Journal · 1970
This paper concerns approximation in the Chebyshev, or minimax sense such that (i) a minimax approximation implies a maximum number of zero error points separated by equal error extrema, and (ii) the approximating function can be so formulated that the disposable parameters are all the coefficients in a polynomial, which may however be part of a more complicated function the rest of which is prescribed. Weighted minimax polynomial approximations can be included, by multiplying the approximated and approximating functions by the weight factor. Analytic methods are described which yield approximately equal error extrema. They are sufficiently simple so that they may sometimes compete with currently used iterative numerical methods, especially when the degree of the disposable polynomial is large. Their most probable utility concerns explorations of available accuracies over wide ranges of design parameters such as degree of disposable polynomial, interval of approximation, and coefficients in prescribed parts of the approximating function.