Hermite-Padé approximation of exponential functions (Q2821896)

Given distinct complex numbers \((\lambda_p)_{p=0}^k\), the (diagonal) Hermite-Padé problem of this paper is to find polynomials \((A_n^p)_{p=0}^k\) of degree at most \(n-1\) and \(A_n^k\) monic such that \(R_n(z):=\sum_{p=0}^k A_n^p(z)e^{\lambda_p z}=O(z^{kn+n-1})\), \(z\to0\). When the \(\lambda_p\) are real, then the polynomials \((A_n^p)_{p=0}^k\) minimize the largest absolute value of \(R(z)\) in the disk \(\{z\in\mathbb{C}: |z|\leq\rho\}\). This minimal value is denoted as \(E_{n-1}(\lambda_1,\ldots,\lambda_k,\rho)\). The main results of this paper include the asymptotics of the residuals \(R_n\) (for complex \(\lambda\)'s) and of the polynomials \(A_n^p\) and corresponding errors \(E_n\) (for real \(\lambda\)'s) as \(n\to\infty\). This generalizes several previous results from the literature e.g. [\textit{P. B. Borwein}, Constr. Approx. 2, 291--302 (1986; Zbl 0625.41012)] and [\textit{F. Wielonsky}, J. Approx. Theory 90, No. 2, 283--298, Art. No. AT963081 (1997; Zbl 0923.41008)], and other related work by Saff, Varga, and Stahl.