Slepian functions on the sphere, generalized Gaussian quadrature rule

Denote by K the operator of 'time-band-time' limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grunbaum et al (1982 SIAM J. Appl. Math. 42 941-55) such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.