Order-constrained uniform approximations to the exponential based on restricted rationals (Q1096822)

The authors [(*) Lect. Notes Math. 1105, 466-476 (1984; Zbl 0578.41019)] considered the restricted-denominator approximations: \[ (1)\quad R_{n/m}(z;\gamma)=\sum^{n}_{i=0}(-1)^ m(\gamma z)^ iL_ m^{(m-i)}(1/\gamma)/(1-\gamma z)^ m \] to the exponential function exp(z), \(z\in C\), where \(L_ m^{(i)}(x)\) is the ith derivative of the Laguerre polynomial (2) \(L_ m(x)=\sum^{m}_{j=0}\frac{(-1)^ j}{j!}\left( \begin{matrix} m\\ j\end{matrix} \right)n^ j.\) For \(\gamma \in R^+\) the order of \(R_{n/m}(z,\gamma)\) as an approximation to the exponential is at least n. Further, the order is \(n+1\) whenever \[ (3)\quad L^{(1)}_{n+1}(1/\gamma)=0\;if\;n=m,\quad =L_ m^{(m-n- 1)}(1/r)=0\;if\;n<m. \] For application purposes to the numerical solution of ordinary differential equations, the authors considered only \(n\leq m\). In the present paper the authors improve their own results and prove some interesting results. Theorem one is a result of (*). In theorem two the authors prove that the global minimum is the least of the local minima and in Theorem three the authors derive the existence of a single local minimum in each interpolation interval, on the line of (*).