Function f(t) specified on [t_0;t_1] has a necessary number of derivatives. Find algorithm which can build uniform approximations of this function with help of partial sums: \sum_{i=1}^{N}\alpha_i e^{-\beta_i t}
That is, find such \alpha_i, Re(\beta_i) \geq 0 satisfying expression:
\min_{\alpha_i, ~\beta_i}\left( \max_{t\in [t_0;t_1]} \left| f(t)-\sum_{i=1}^{N}\alpha_i e^{-\beta_i t}\right| \right)

Also consider a function specified on [t_0;+\infty)

Thanks for any ideas!