Spherical Zone T-designs for Numerical Integration and Approximation

In this paper, we present spherical zone t-designs, which provide quadrature rules with equal weight for spherical polynomials of degree at most t on a spherical zone {X is an element of S² :cos theta <= x. z <= cos theta-} with z is an element of S² and 0 <=theta-<theta <=π. The spherical zone t-design is constructed by combining spherical t-designs and trapezoidal rules on [0,2π] with polynomial exactness t. We show that the spherical zone t-designs using spherical t-designs only provide quadrature rules with equal weight for spherical zonal polynomials of degree at most t on the spherical zone. We apply the proposed spherical zone t-designs to numerical integration, hyperinterpolation and sparse approximation on the spherical zone. Theoretical approximation error bounds are presented. Some numerical examples are given to illustrate the theoretical results and show the efficiency of the proposed spherical zone t-designs.