I have some scientific code which interfaces with a library which accepts real functions specified as any number of additive terms with exponential powers. For instance, it is capable of accepting function
```
0.1 x^(3.14) - 0.2 x^(-156.4)
```
where every coefficient and exponent can be specifiable up to machine epsilon.
I need to approximate a tractable sum of Gaussian functions into this form, to dispatch them to the library with tenable accuracy.

For my purposes, a sufficiently accurate Taylor expansion required `60` terms with powers as large (in magnitude) as `69`, and a similar diagonal Padé approximant required `20` terms with powers up to `18`. But since these powers are restrictedly integers, I cannot help but wonder whether a smaller and more accurate expansion is possible with permittedly rational powers.

I would imagine the uncountably infinite family of rational-power expressions do not satisfy the necessary conditions (like orthonormality, clearly) to constitute a basis, and hence will not admit an analytic prescription for determining the coefficients. I would further imagine that a practical way to find coefficients and powers which produce a sufficiently accurate approximation in a given region is best done with some numerical/iterative optimisation. I am happy to do this, but the expression (like the example above) must be performed in advance, independent of any `x` value.

My question:

 - is my intution correct that sums of real-weighted rational powers (bounded, let's say, in `[-20,20]`) can "better" approximate functions like gaussians (in some regime about zero)?
 - is there a protocol (which needn't be efficient) to well approximate the conditions?