We know that a function $f(t)$ has the form $$ f(t)=x_1 e^{z_1 t}+x_2 e^{z_2 t}+\cdots+x_n e^{z_n t}, $$ for some unknowns $x_i, z_i$ but we can calculate the value $f(t)$ for any $t.$
Question. How to define the $x_1,x_2,\ldots, x_n, z_1, z_2, \ldots z_i$ in terms of the values $f(t)$?
My attempt. Put $y_i=e^{i z_1}.$ Then we get the following system of non-linear equations \begin{cases} x_1+x_2+\cdots+x_n=f(0),\\ x_1 y_1+x_2 y_2+\cdots+x_n y_n=f(1),\\ x_1 y_1^2+x_2 y_2^2+\cdots+x_n y_n^2=f(2),\\ \ldots \\ x_1 y_1^{2n-1}+x_2 y_2^{2n-1}+\cdots+x_n y_n^{2n-1}=f(2n-1). \end{cases}
Is there a nice general solution of the system?