2000•Electronics and Communications in Japan (Part III Fundamental Electronic Science)Requires access
On the relation between the maximum errors of the leastp-th approximation and those of the minimax approximation by a rational function
Tetsuo Nishi, Feng Lü
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Abstract
The minimax approximation is of special importance for the design of filters. As to the minimax approximation by a rational function, the characteristics of the best approximations are well developed. However, least-squares approximation is often used instead due to the simplicity of calculation. Nevertheless, no theoretical result has been published as to how close and how good the achieved approximation results are compared to those of the best (optimal) minimax approximation. This paper deals with the least p-th approximation (p ⩾2, even) by a rational function and gives a fairly simple theoretical lower bound for the ratio of the maximum error(s) of the minimax approximation to those of the least p-th approximation (usually local minima). This paper proves that on a weak but practical assumption, Nishi's result on the least p-th approximation by linear function is also valid for the approximation by rational function. Numerical examples show that the least p-th approximation for p = 8 or 16 is usually enough to achieve a good approximation to the minimax approximation. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(3): 21–32, 2001