Rational approximation to ex and to related functions

AbstractAccording to a well-known result of S. N. Bernstein, ex can be approximated uniformly on [−1, 1] by polynomials of degree ⩽n with an error of the order [2n(n + 1)!]−1. In this note it is shown that the smallest (uniform norm) error in approximating ex by reciprocals of polynomials of degree ⩽n is also of the order [2n(n + 1)!]−1. We denote throughout by Pn(x), Qn(x) real polynomials of degree ⩽n. We show, furthermore, that the smallest error in approximating ex by rational functions of the form Pn(x)Qn(x) where the coefficients of Qn are ⩾0 is again of that same order