CUBATURE FOR THE SPHERE AND THE DISCRETE SPHERICAL HARMONIC TRANSFORM

摘要

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 U-n cannot achieve the classical lower bound of dimP(s), where P-s is the space of ail polynomials in n variables of at most degree s restricted to U-n. 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. [References: 19]