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For an application I have to approximate a continuous (and hopefully smooth) positive even function that decays at infinity with a sum of sums of gaussians, preferably orthogonal ones.

That is, a function $$f(x)|f(x)=f(-x),\lim_{x\to \infty}f(x)=0$$ I'd like to approximate with $$\Sigma_ng_n(x)|g_n(x)=\Sigma_ka_{nk}e^{-(x/s_{nk})^2}$$ so that $$\int g_ng_m=\delta_{mn}$$ and $$lim_{n\to\infty}\Sigma_ng_n(x)=f(x)$$

In addition, I would prefer for each $g_n$ to have bounded, eh, "conditioning number": $$\forall n \frac{\max_k(s_{nk})}{\min_k(s_{nk})}<C$$ where $C$ does not depend on $n$.


One way to go about it is to set $$y=e^{-x^2}$$ then approximate $$h(y):=f(x)$$ with Chebyshev polynomials over $y$, and then change variables back.

However, I don't like this solution very much because it limits $s_{nk}$ to square roots of integers and therefore unlikely to be optimal. In addition, the above "conditional number" won't be bounded.


Since I'm doing this for an application the speed of convergence matters very much. Could you suggest another method for producing the above approximation or a reference that considers such problems?


EDIT: the above seems to be related to Radial basis function approximations.

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    $\begingroup$ How can gaussians, being positive nonvanishing functions, be orthogonal? $\endgroup$ – Igor Rivin Jan 10 '17 at 2:49
  • $\begingroup$ @IgorRivin, Gaussian cannot be, but $g_n$ are sums of Gaussian, with coefficients $a_{nk}$ that don't have to be nonnegative. $\endgroup$ – Michael Jan 10 '17 at 5:01
  • $\begingroup$ The span of Gaussians is dense in the space of even Schwartz functions on $\mathbb{R}$. Density refers to the usual topology on the space of Scwartz functions. See Lemma 2.2 in unc.edu/math/Faculty/met/bessel2.pdf $\endgroup$ – Liviu Nicolaescu Jan 10 '17 at 9:54
  • $\begingroup$ Ah, I misunderstood the question somewhat... $\endgroup$ – Igor Rivin Jan 10 '17 at 15:52
  • $\begingroup$ Liviu, that link no longer works and I couldn't dig up the pdf by Googling. $\endgroup$ – Jess Riedel Sep 13 '20 at 18:15
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Approximate a function with a sum of sums of gaussians, preferably orthogonal ones. ... the above seems to be related to Radial basis function approximations.

Indeed, you could use an RBF interpolant at chosen centers as an approximation for your function. Then apply a Gram–Schmidt orthogonalization to construct linear combinations of the RBFs that are orthogonal with respect to a given inner product.

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