# Optimal orthonormal basis to represent a Gaussian

I am looking at representing a set of Gaussians, of the form $\exp(-\frac{(r-r_i)^2}{2 \sigma^2})$, on a 1D domain. I do not know $r_i$ and $\sigma$ prior to defining the basis $\{ \phi_k(r) \}_{k=1}^n$. For the representation, I want to use a basis which can be defined based on four criteria:

1. The basis is as complete as possible given a target number $n$ of basis functions.
2. The basis functions are fully defined from the domain size, $r \in (0,r_\text{cut})$, and $n$. That is, the basis functions depend on $r$ and, parametrically, on $r_\text{cut}$ and $n$.
3. The basis is orthonormal.
4. The basis is optimally suited for representation of Gaussian functions. This means that I can obtain the expansion coefficients analytically.

Basically, the end result is an approximant to my original function:

$\exp(-\frac{(r-r_i)^2}{2 \sigma^2}) \approx \sum\limits_{k=1}^n w_k \phi_k (r)$

where, as said, I'm aiming at being able to obtain the $w_k$ analytically. These expansion coefficients can depend parametrically on $r_i$, $\sigma$, $r_\text{cut}$ and $n$. Is there any basis suited for this problem?

Why I want to do this

My final application requires a 3D representation of a series of points. The radial representation is done using Gaussians along the radial direction; the angular representation is done using spherical harmonics. The whole goal of doing a representation in a basis is to use the expansion coefficients. These expansion coefficients can be used to build a rotationally-invariant discrete representation of the set of points. For this to be computationally efficient, the basis needs to be as small as possible. To represent the radial part, I want a fixed basis which is optimally suited to represent 1D Gaussians. I cannot use the Gaussians themselves because I need to represent literally millions of Gaussians using a fixed-size basis, which needs to be always the same.

• Gaussian has a converging taylor series expansion. So any polynomial basis such as Lagrange or Legendre polynomials should work. Note that representing gaussians with very narrow width is a notoriously difficult problem though. If the gaussian is allowed to be very narrow and be located anywhere in the bounded domain, you need a basis that can be arbitrarily sharp in any part of the domain, and that part is not known a priori, so an optimal basis does not exist. If you have bounds on $\sigma$, you can calculate how many basis functions you will need for a given error – Aleksejs Fomins Aug 23 '18 at 11:25
• Thanks, @AleksejsFomins, that provides me with a good basis (pun intended) to work with. I actually started working with some regular polynomials to get going, but an optimally efficient basis further down the road may be required. – Miguel Aug 23 '18 at 11:34
• Thing is, if you want to calculate the coefficients analytically, you are in luck with polynomials, because integral of polynomial * gaussian can be written in terms of error functions analytically. Most other basis like fourier can actually be expressed in terms of a polynomial basis, which means that they will offer the same accuracy. If you choose to go into wavelets or other crazy stuff, I'm not sure if it is going to be better, but most likely you will not be able to get analytical expressions for your coefficients. I would be surprised if there is something easily better than polynomials – Aleksejs Fomins Aug 23 '18 at 11:38
• I'd be interested to know what advantage working with a basis brings to your application that cannot be obtained by using the Gaussian functions directly. These have an explicit formula and converge rapidly to zero away from the mean, and in 1D I'd expect the bookkeeping to be rather compact. – hardmath Aug 23 '18 at 16:10
• But I'm responding to your Question as written, which describes a need for "representing a set of Gaussians.. on a 1D domain." If there is some difficulty in just using Gaussians, you will probably need to share more context to get help with that. Piecewise polynomials, eg. B-splines, are often useful to trade-off compact support for some loss of smoothness, etc. But with no information what your basis will be used for, I can't give much value in advice. – hardmath Aug 23 '18 at 17:03