Supplementary Material of Rational neural networks

Thus, xr(x) is a rational approximant to |x| of type at most (k + 1, k). Let 0 < l < 1 be a real number and consider the sign function on the domain [ 1, l] [l, 1], i.e., We refer to such r(x) as the Zolotarev sign function. Moreover, since xr(x) 0 for x [ 1, 1] (see [2, Equation (12)]) we have max ||x| xr(x)| max |x| l. One finds that l = 4 exp( π k/2) and the result follows immediately. The proof of Lemma 1 is a direct consequence of the previous lemma and the properties of Zolotarev sign functions.