  Positive definite functions of compact support are widely used for radial
basis function approximation as well as for estimation of spatial processes in
geostatistics. Several constructions of such functions for ${\mathbb R}^d$ are
based upon recurrence operators. These map functions of such type in a given
space dimension onto similar ones in a space of lower or higher dimension. We
provide analogs of these dimension hopping operators for positive definite, and
strictly positive definite, zonal (radial) functions on the sphere. These
operators are then used to provide new families of strictly positive definite
functions with local support on the sphere.
