On the relation between the maximum errors of the least p-th approximationand those of the minimax approximation by a rational function

The minimax approximation is of special importance for the design of filter s. As to the minimax approximation by a rational function, the characterist ics of the best approximations are well developed. However, least-squares a pproximation is often used instead due to the simplicity of calculation. Ne vertheless, no theoretical result has been published as to how close and ho w good the achieved approximation results are compared to those of the best (optimal) minimax approximation. This paper deals with the least p-th appr oximation (p greater than or equal to 2, even) by a rational function and g ives a fairly simple theoretical lower bound for the ratio of the maximum e rror(s) of the minimax approximation to those of the least p-th approximati on (usually local minima). This paper proves that on a weak but practical a ssumption, Nishi's result on the least p-th approximation by linear functio n is also valid for the approximation by rational function. Numerical examp les show that the least p-th approximation for p = 8 or 16 is usually enoug h to achieve a good approximation to the minimax approximation. (C) 2000 Sc ripta Technica.