Series expansion for gaussian-like function

Question

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?

Answer 1

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.