Does anyone know of any references for finding the minimiax polynomial approximation to x^N on some interval that has degree k < N?
Google for Remez algorithm. And maybe "equal ripple". I wrote one in Macsyma for Livermore, but I forget which book I cribbed from. The idea is brilliantly simple. It's a quadratically convergent iteration that alternately equalizes amplitudes at the estimated turning points, and then recomputes those turning points. It works equally well for rational functions, and my Macsyma program will try to determine coefficients in just about any smooth approximating expression. But in practice, as soon as you get a little bit fancy, locating, and even counting the turning points gets hairy, and you need some way to give the program advice. E.g., your optimal solution may have more ripples than degrees of freedom, the surplus of which are of lesser amplitude. This can even happen with polynomial approximations if the coefficents are interdependent. Also, you typically need multiprecision numerics. (And, of course, symbolic differentiation and Newton's method or the like.) --rwg ARTHROSCOPE PROTHORACES CRAPSHOOTER