.. _DefSmoothingKernel:

Definition of term: Smoothing kernel
====================================

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

#. :math:`\int_{-\infty}^{+\infty}\kappa(u)\,du` is finite

#. :math:`\kappa(-u) = \kappa(u)` for all values of :math:`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
:ref:`AltMultivariateCovarianceStructures<AltMultivariateCovarianceStructures>`),
in which case they depend on some hyperparameters. An example of a
smoothing kernel in this context is

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

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