Uniform Approximation Algorithm for Function f(t) on [t_0;t_1] with Partial Sums

[Tolya]

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 the 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!

 

[lippyka]

For a function specified on [t_0;t_1], one approach is to use a least squares approximation. The idea is to find the coefficients \alpha_i, \beta_i that minimize the sum of squared errors between the function and its approximation:\sum_{i=1}^{N} \left( f(t_i) - \sum_{j=1}^{N}\alpha_j e^{-\beta_j t_i} \right)^2where t_i are a set of points in [t_0;t_1]. This can be done by solving a linear system of equations or using optimization algorithms such as gradient descent.For a function specified on [t_0;+\infty), a solution could be to use an exponential series expansion. That is, approximate the function with a sum of exponentials of the form:\sum_{i=1}^{N}\alpha_i e^{-\beta_i t},where the coefficients \alpha_i and \beta_i are found by solving a linear system of equations. This can be done by expanding the function as a Taylor series and then truncating the series at some order.