I need a series expansion to describe a general gaussian-like (bell shaped) function. I couldn't find a rigorous definition of "bell shaped" online but in essence the function should have the following:


1. $f(x)>0$ (positive)
2. $f(x)=f(-x)$ (symmetric)
3. $\int \_{-\infty}^{\infty}f(x)dx <\infty$
4. $f^{(n)}(x) = 0$ has exactly $n$ roots (bell shaped)
5. $f^{(n)}(x) \rightarrow 0 $ as $|x| \rightarrow \infty$ for all $n$


Example would be: Gaussian, Lorentzian curve, Voigt profile, Sech.


I need a series expansion that the gives approximants for all such functions. The series expansion should only give "bell shaped" functions for any choice coefficients (as with fourier series - for example - only producing periodic functions).


Does this exist?



Series expansion for gaussian-like function
You could try a series of Hermite functions,
$$f(x)=\sum\_{n=0}^N a\_n \frac{d^{2n}}{dx^{2n}}e^{-x^2}.$$
The function $f$ satisfies your conditions 1,2,3,5 by construction, and if the $a\_n$'s decay rapidly with $n$ it will look "bell-shaped".


Actually, for unconstrained $a\_n$'s, and including also odd derivatives, this expansion is dense in $L^2(\mathbb{R})$, see [Approximating with Gaussians.](https://arxiv.org/abs/0805.3795)


