Rational approximation of 𝐱ⁿ

Abstract

Let E k k ( n ) E_{kk}^{(n)} denote the minimax (i.e., best supremum norm) error in approximation of x n x^n on [ 0 , 1 ] [0,1] by rational functions of type ( k , k ) (k,k) with k > n k>n . We show that in an appropriate limit E k k ( n ) ∼ 2 H k + 1 / 2 E_{kk}^{(n)} \sim 2 H^{k+1/2} independently of n n , where H ≈ 1 / 9.28903 H \approx 1/9.28903 is Halphen’s constant. This is the same formula as for minimax approximation of e x e^x on ( − ∞ , 0 ] (-\infty ,0] .