Hermitian approximation of two exponents

We study the asymptotic properties of diagonal Pade–Hermite approximants $\{\pi^{j}_{2n,2n}(z;e^{j\xi;})\}^{2}_{j=1}$ for a system consisting of functions $\{e^z,e^{2 z}\}$. In particular, we determine the asymptotic behavior of the differences $e^{jz} - \pi^j_{2n,2n}(z; e^{j\xi})$ for $j =1,2$ and $n \to\infty$ for any complex number $z$. The obtained results supplement research of Pade, Perron, Braess and A.I. Aptekarev dealing with the study of the convergence of joint Pade approximants for systems of exponents.