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.