Definition of term: Smoothing kernel

A smoothing kernel is a non-negative real-valued integrable function \(\kappa()\) satisfying the following two requirements:

1. \(\int_{-\infty}^{+\infty}\kappa(u)\,du\) is finite
2. \(\kappa(-u) = \kappa(u)\) for all values of \(u\)

In other words, any scalar multiple of a symmetric probability density function constitutes a smoothing kernel.

Smoothing kernels are used in constructing multivariate covariance functions (as discussed in AltMultivariateCovarianceStructures), in which case they depend on some hyperparameters. An example of a smoothing kernel in this context is

\[\kappa(x)=\exp\{-0.5 \sum_{i=1}^p (x_i/\delta_i)^2 \} \, ,\]

where \(p\) is the length of the vector \(x\). In this case the hyperparameters are \(\delta=(\delta_1,...,\delta_p)\).