In generalized hyperinterpolation on the sphere, the
Newman-Shapiro kernels are the favourable tool, with outstanding properties.
The complexity of this approximation method can be reduced considerably by
truncation. In this case, kernel functions are used, only, whose
pole is located rather near to the evaluation point. So the question
arises, how the kernels, which are zonal spherical polynomials, behave near
their pole for increasing degrees. We answer this question by the help of
certain Bessel-functions, and discuss some implications in truncated
generalized hyperinterpolation.