Product of Gaussian functions

Main reference: Products and Convolutions of Gaussian Probability Density Functions

Contents:

1. Product
2. Inner product
3. Kernel operator

Consider the the Gaussian-like function
$$g(x) = \exp{\frac{-(x-p)^2}{w^2}}$$
where $p$ and $w$ stand for position and (half) width of $g$.

Note that, see e.g. MSE,
$$\int_{\mathbb R} e^{-x^2}dx=\sqrt{\pi}.$$
Thus,
$$\int_{\mathbb R} g(x)dx=w\sqrt{\pi}.$$

1. Product

Observation: The product of two Gaussian functions is again a scaled-Gaussian function. In particular, the scaling factor is also a Gaussian.

We have
$$A:=\frac{(x-p_1)^2}{w_1^2}+\frac{(x-p_2)^2}{w_2^2}=\frac{w_1^2+w_2^2}{w_1^2w_2^2}(x-\frac{w_2^2p_1+w_1^2p_2}{w_1^2+w_2^2})^2+ \frac{(p_1-p_2)^2}{w_1^2+w_2^2}$$
Let $p_{12}$ and $w_{12}$ defined by the following relations

$$\frac{1}{w_{12}^2}=\frac{1}{w_{1}^2}+\frac{1}{w_{2}^2} \text{ and } \frac{p_{12}}{w_{12}^2}=\frac{p_1}{w_{1}^2}+\frac{p_2}{w_{2}^2}$$

Thus,
$$g_1(x)g_2(x)=\exp(-A) = \exp\frac{-(p_1-p_{2})^2}{w_{1}^2+w_2^2} \exp\frac{-(x-p_{12})^2}{w_{12}^2}.$$

2. Inner product

$$\langle g_1, g_2\rangle =\int_{x\in \mathbb R} g_1(x)g_2(x) dx = \sqrt{\pi} w_{12} \exp\frac{-(p_1-p_{2})^2}{w_{1}^2+w_2^2}.$$

Observations:

1. Fix $p_1, w_1$, let $p_2=p$, $w_2=w$ be varying. Then this inner product is maximized if $p=p_1$ and $\dfrac{1}{w_{12}^2}$ is minimized, i.e. $w$ is maximized.
2. $||g||^2 = w \sqrt{\pi/2}$ or $\dfrac{1}{||g||}=\sqrt[4]{\dfrac{2}{\pi}}w^{-1/2}$

3. Kernel

Let $t=(p, w)$ and $a_t(x)=\dfrac{g}{||g||}$. Define
$$K(t_1, t_2) = \int_{x\in \mathbb{R}} a_{t_1}(x) a_{t_2}(x)dx=\langle a_{t_1}, a_{t_2}\rangle\leq 1.$$

Observation: The kernel $K$ achieves its maximum value being equal to $1$ iff $t_1=t_2$. Indeed,
$$K(t_1, t_2) =\sqrt{\frac{2}{\pi}} w_1^{-1/2} w_2^{-1/2} \sqrt{\pi} w_{12} \exp\frac{-(p_1-p_{2})^2}{w_{1}^2+w_2^2}.$$
or
$$K(t_1, t_2) =\sqrt{\frac{2w_1w_2}{w_1^2+w_2^2}} \exp\frac{-(p_1-p_{2})^2}{w_{1}^2+w_2^2}.$$
which equals to $1$ iff $w_1=w_2$ and $p_1=p_2$.