A pole problem in the reduced Gaussian grid

It is shown that a pole problem may arise from the use of a reduced Gaussian grid in the spectral transform method on the sphere. The problem is related to an asymptotic property of the associated Legendre functions and can be solved by slightly increasing the number of points close to the pole. It is also shown that the reduced grid controls aliasing arising from quadratic terms only as an asymptotic property. Nevertheless, a small increase in the number of points (everywhere) is enough to reduce the aliasing to a negligible level.