Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? - MathOverflow

Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?

2012-02-12T11:42:07Z

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.

Answer by psd for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?

2012-02-12T22:51:39Z

It could be cleaner to use a mixture of beta distributions rather than of truncated normal distributions if you want to approximate an arbitrary distribution on an interval.

From this paper:

Although the GMM (Gaussian mixture model) can model arbitrary distributions with a proper number of mixture components, a large number of these components are spent on describing the edge when modeling semibounded or bounded support data. Compared to the Gaussian distribution, the beta distribution has a more flexible shape. It has a support range of $[0,1]$, and can be easily generalized to any compact range $[a,b], a, b \in \mathbb{R}$.

Answer by Zander for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?

2012-02-13T01:06:33Z

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). So you can use 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.

Answer by Dirk for Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes?

2012-02-13T11:31:46Z

While @Zander is right about the fact that "these functions cannot make a basis for the functions space" there are still a lot of families of translates of "bell shaped functions" which for a dense set in $L^2$. You may consider the paper "On aproximations by shifts of the Gaussian function" which treats exactly this problem.

Indeed, one knows that for the Gaussians $\phi(x) = \exp(-\pi x^2)$ it holds that the span of the functions $\phi(x-\lambda_k)$ is dense in $L^2(\mathbb{R})$ if the shifts $\lambda_k$ fulfill that the sum $\sum_k \lambda_k^{-2}$ diverges - imho a quite surprising result...