I am trying to "model" Fig 2 with a superposition of a bump function. I understand that bump functions are bounded and can be often differentiated. The bump function I have used is shown in Fig 1.
My bump function:
$$f(x)=exp\left(\frac{x^2}{x^2-L^2}\right)$$
$L$ varies from $-\pi$ to $\pi$ and produces the plot below (FIG 1):
FIG 2What I am trying to model through a superposition of such bump functions ALSO AVAILABLE HERE:
What kind of superposition of bump functions will I need? I have tried fractional additive superpositions such as:
$$f(x)=exp\left(\frac{x^2}{x^2-L^2}\right) +0.01 exp\left(\frac{(x+2)^2}{x^2-L^2}\right)$$
$\ldots$ but that didn't work!
EDIT 1: Per fgp's comment below:
$L$ is a constant. In my plot $x$ ranges between $-L$ to $L$. I used Mathematica to do this plot and the code, in case anyone is interested is:
L=\[Pi];
f= Exp[x^2/(x^2-L^2)]
Plot[
f,
{x,-L,L}
]
EDIT 2: In an attempt to "scale and translate" as per Henning Malkholm, I produced this monstrosity which is not quite correct
$$a \exp \left(\frac{(2 x+3 \pi )^2}{(2 x+3 \pi )^2-(3 \pi )^2}\right)+a \exp \left(\frac{(2 x-3 \pi )^2}{(2 x-3 \pi )^2-(3 \pi )^2}\right)+\exp \left(\frac{x^2}{x^2-L^2}\right)$$
The scale factor a did not make too much of a difference as a number. I suppose I would need a function a=a(x) to do this.
Any ideas at this stage would be appreciated.
How about just scaling and translating?
$$ g(x) = f(x) + af(2x-3\pi) + af(2x+3\pi)$$ where $f$ is your original bump function, for some appropriate coefficient $a$.
If that "doesn't work", you'll need to define "works" more clearly and tell us how it doesn't work.
Edit after question was updated: I see what's happening. Your bump function should actually be
$$ f(x) = \begin{cases} \exp\left(\frac{x^2}{x^2-L^2} \right) & \text{when } |x|<L \\ 0 & \text{when } |x|\ge L\end{cases} $$
Outside the interval $(-L,L)$ the first expression will give you large numbers that are not part of the bump, so you need a piecewise definition to get rid of them.