The Gaussian kernel of bandwidth 
on Euclidean space
is the radial function
![K_sigma(x,y)=exp[-(||x-y||^2)/(2sigma^2)].](/images/equations/GaussianKernel/NumberedEquation1.svg) |
It is a positive-definite kernel, meaning that its Gram matrix on every finite point set is positive
semidefinite. It is also a Gaussian radial
basis function. Multiplying by 
gives a normalized Gaussian density in 
dimensions, the form used as a kernel
in kernel density estimation.
The same Gaussian shape, with a different parameterization and normalization, is the fundamental solution of the heat equation. Thus the phrase Gaussian kernel is used in several related settings, and the scale convention must be specified.
