  As to the Bessel integrals of type \begin{equation*} \int_0^x
\left(x^\mu-t^\mu\right)^\lambda t^\alpha J_\beta(t)dt\qquad(x>0),
\end{equation*} we improve known positivity results by making use of new
positivity criteria for ${}_1F_2$ and ${}_2F_3$ generalized hypergeometric
functions. As an application, we extend Buhmann's class of compactly supported
radial basis functions.
