Cubature for the Sphere and the Discrete Spherical Harmonic Transform

Using a result of Bannai and Damerell, it is shown that a cubature formula with N points of degree 2s > 4 for the surface of the n-dimensional sphere Uₙ cannot achieve the classical lower bound of Pˢ, where Pˢ is the space of all polynomials in n variables of at most degree s restricted to Uₙ. This implies that for n > 2 there does not exist a cubature-based discrete n-dimensional spherical harmonic transform for degree s > 2 with the same number of points as spectral coefficients.