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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. 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?
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  • $\begingroup$ I am confused by the fact that you can feed the library Padé approximants. They don't seem to be in the acceptable form, because they have poles in places other than zero. What am I missing? $\endgroup$
    – Federico Poloni
    Commented Aug 23, 2022 at 12:48
  • $\begingroup$ @FedericoPoloni I feed diagonal Padé approximants using this Mathematica function (second overload) $\endgroup$
    – Anti Earth
    Commented Aug 23, 2022 at 13:05
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    $\begingroup$ Exactly -- how is something like $\frac{1+\frac{x}2 + \frac{x^2}{12}}{1-\frac{x}2 + \frac{x^2}{12}}$, which is the $(2,2)$ Padé approximant to $\exp(x)$, in that format? $\endgroup$
    – Federico Poloni
    Commented Aug 23, 2022 at 13:24
  • $\begingroup$ @FedericoPoloni ah you helped me find a very unexpected bug in my notebook haha, thank you! Scratch my every mention of Padé approximants $\endgroup$
    – Anti Earth
    Commented Aug 23, 2022 at 13:52
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    $\begingroup$ If you're asking about approximating a function by sums of rational powers, you'll want to have a look at Puiseux series. Whether or not this gives you some advantage for approximating a particular function like a Gaussian is another question; I have a hunch that you'll do better with the same number of coefficients using Padé but that's just a guess. $\endgroup$
    – Daniel Shapero
    Commented Aug 23, 2022 at 21:57

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