The problem is that we want to approximate domain-bounded functions (that is, functions restricted to a domain such as [0,10]) as (probably infinite) series of other functions. We know that, for integrable functions in the interval, we can definitely do that with sin waves (that's the fourier transform), but we wondered if we could use other kind of functions as well. In particular, we are wondering if we could use for the series a bell-shaped function like the following:


$fbell(s,c,h,r)=\begin{cases}
h-2h\frac{\parallel s-c\parallel^{2}}{r^{2}} & if \parallel s-c\parallel\lt r/2 \end{cases}$


$fbell(s,c,h,r)=\begin{cases} \frac{2h}{r^{2}}(\parallel s-c\parallel-r)^{2} & ifr\gt\parallel s-c\parallel\geq r/2 \end{cases}$


$fbell(s,c,h,r)= \begin{cases}0 & \parallel s-c\parallel\geq r\end{cases}$


where c is the center of the bell, h is the weight and r is the radius. If I'm not mistaken, t is continuous and differentiable in $\parallel s-c\parallel\lt r$.


It is useful for us because we work in utility theory and this function makes sense where there is a set of utility peaks and utility fades as we go far from them.


I understand that this is related to the Wavelet transform and the orthonormal wavelets, but I'm not sure this is fully applicable to our case (where the domain of the function is bounded). 


Thanks in advance.



Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?
You can't approximate arbitrary square-integrable functions in the same way as a Fourier series or wavelet series, since your functions cannot make a basis for the function space.


But what you described is a quadratic spline (almost but not quite a usual [basis B-spline](http://en.wikipedia.org/wiki/B-spline)). So you can use [spline interpolation](http://en.wikipedia.org/wiki/Spline_interpolation) to approximate an arbitrary function $f$ in the sense that you can pick a grid of "knots" and find a set of $h$ weights so that the sum of your $fbell$ splines is a function that equals $f$ at each of the knot points and has a continuous first derivative. If you have almost any kind of smoothness constraint on $f$ (e.g. a bound on some derivative), that will allow you to approximate $f$ to within any specification by choosing a sufficiently fine grid of knots.


