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

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

Thanks for any ideas!