So I have a small list of pairs of the form $(\mathbb{R}, \mathbb{R}_{\geq 0})$ and I want to fit a function to this data. Additionally I know that as $x$ grows large in either direction $y$ tends to zero (in fact I know that $y$ approaches zero at $\infty$ and $-\infty$). Additionally $y$ never approaches $\infty$ There may be many peaks in the middle as well. More generally I actually need to be able to fit data of the form $(\mathbb{R}^n, \mathbb{R}_{\geq 0})$ for any $n\in\mathbb{N}$. It should be noted that I am using a program to to fit the function so this need not be easy to do by hand. My first thought was to use a polynomial but polynomials do not tend to zero. My next thought was a rational function with a denominator larger than denominator but then there will be many vertical asymptotes giving values of y that approach $\infty$.
Do you know of such a form of a function that meets the above requirements (y never approaches infinity, y approaches zero at extreme, y is always positive, the function may have many peaks) that I can fit the constants of? (you may assume that I can fit any constants no matter what).
edit: To be clear here are the requirements formally stated of $f(x)$ (a function which could model data)
- $lim_{x\rightarrow \infty}f(x)=0$
- $lim_{x\rightarrow -\infty}f(x)=0$
- $\frac{d}{dx}(f(x)) = 0$ should have more than one solution
- $\forall x\in \mathbb{R}. f(x) \ge 0$
- $\forall a\in \mathbb{R}. lim_{x\rightarrow a}f(x) \ne \infty$
- The function should have unspecified constants that I can fit to my data
- It is allowed that there be some requirements on the allowed values of the constants