  Kernel methods have been widely applied to machine learning and other
questions of approximating an unknown function from its finite sample data. To
ensure arbitrary accuracy of such approximation, various denseness conditions
are imposed on the selected kernel. This note contributes to the study of
universal, characteristic, and $C_0$-universal kernels. We first give simple
and direct description of the difference and relation among these three kinds
of universalities of kernels. We then focus on translation-invariant and
weighted polynomial kernels. A simple and shorter proof of the known
characterization of characteristic translation-invariant kernels will be
presented. The main purpose of the note is to give a delicate discussion on the
universalities of weighted polynomial kernels.
