Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles

We describe an explicit construction of quadrature rules exact for integrating multivariate trigonometric polynomials of a given coordinatewise degree on a spherical triangle. The theory is presented in the more general setting of quadrature formulas on a compact subset of the unit hypersphere, S ^q, embedded in the Euclidean space R ^q+1. The number of points at which the polynomials are sampled is commensurate with the dimension of the polynomial space.