Approximating the pth root by composite rational functions

A landmark result from rational approximation theory states that x
1/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
x
1/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 x
1/p
and related functions (such as |x| and the
sector function) with exceptional efficiency