Approximating the [formula omitted]th root by composite rational functions

Abstract

A landmark result from rational approximation theory states that x1∕p on [0,1] can be approximated by a type-(n,n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x1∕p by composite rational functions of the form rk(x,rk−1(x,rk−2(⋯(x,r1(x,1))))). While this class of rational functions ceases to contain the minimax (best) approximant for p≥3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x1∕p and related functions (such as |x| and the sector function) with exceptional efficiency.