The Epanechnikov Density is an example of a kernel that has zero contribution outside a range but is smooth inside that range, first suggested by Epanechnikov (1969).
Where is the volume of the dimensional unit sphere6.
So for a 1D KDE, and
and for a 2D KDE, giving
If the density estimate is required for some application where it is only required to proportionality (such as contour plotting) then the following kernel may suffice:
Such kernels save the necessity of determining which points have sufficient influence to be included in the summation, but have some unfortunate mathematical properties that make them unsuitable for the Bayesian KDE in the next section, the main disadvantage being the introduction of an ``edge'' wherever the density arrives at the zero crossing -- if the resulting density is to be smooth, introducing arbitrary cut-offs should be avoided.