ZERO AND POLE DISTRIBUTION OF DIAGONAL PADÉ APPROXIMANTS TO THE EXPONENTIAL FUNCTION

Abstract

The polynomials P_n and Q_m , having degrees n and m respectively, with P_n monic, that solve the approximation problem will be investigated for their asymptotic behaviour, in particular in connection with the distribution of their zeros. The symbol O means that the left-hand side should vanish at the origin at least to the order n+m+1. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main apprcudmants. We shall focus on the important diagonal case when n = m and the polynomials P_n and Q_m , as well as the remainder E_n,n ,(z) = P_n (z)e^−z + Q_n (z) can be expressed in terms of Hankel and Bessel functions. The approximate location of the zeros of P_n , Q_m and E_n,n are given in terms of the known zeros of certain Airy functions. An application is given in which the asymptotic information on the zeros is used to obtain an estimate in an approximation of the unit block function by means of the polynomials P_n , Q_n .