An extension of the `1/9'-problem
Let \(u_0\) and \(u_1\) be rational functions. The authors considers rational approximation of the function \[ f(z)=u_0(z)+u_1(z)e^z \] on the negative half-axis. The main result is that \[ \lim_{n\to\infty}\|f-r^*_{n,n+k}\|^{1/n}=\frac{1}{9.28902549...}=H, \] where \(H\) is the Halpern constant from the solution of the `\(1/9\)'-problem by \textit{A. A. Gonchar} and \textit{E. A. Rakhmanov} [Math. USSR, Sb. 62, No. 2, 305--348 (1989); translation from Mat. Sb., Nov. Ser. 134 (176), No. 3(11), 306--352 (1987 Zbl 0663.30039)].
scientific article; zbMATH DE number 5624895
scientific article