Approximate the sum of exponential functions

Question

I'm trying to approximate the sum of $N$ exponential functions as one function.

Is it possible to do so, i.e. find function $f$ and constants $a$, $b$, and $c$ such that $$\sum_{k=1}^N c_ke^{a_kx}\approx c f(ax+b)$$

Answer 1

It is unlikely that you will be able to approximate it if you look at it as a sum of the exponential function. Replace $x=\ln(y)$ and have an exponential polynomial instead.

$$\sum_{k=1}^N c_ky^{a_k}$$

Since you want an approximation, you can find the reasonable fractional approximation for all $c_k \approx \frac{r_q}{t_q}$ and $a_k \approx \frac{p_k}{q_k}$. Assume that the least common multiple of all $q_k$ is $s$, it would be then $a_k=\frac{m_k}{s}$. Do the same for $t_q$ and find that the common multiple is w making $c_k=\frac{u_k}{w}$

Now replace

$$z=y^{\frac1{s}}$$

$$\frac{1}{w}\sum_{k=1}^N u_kz^{m_k}$$

And $P(z)=\sum_{k=1}^N u_kz^{m_k}$ is a plain integer polynomial.

Other adopted values depend on the precision you want or need.