Quadrature rules with neighborhood of spherical designs on the two-sphere

In this paper, we concentrate on quadrature rules with their point sets located on a neighborhood of a spherical design. We show that any point set in a small enough neighborhood of a fundamental spherical design can establish a positive quadrature rule. A preliminary bound of the neighborhood radius is given to guarantee this property. The perturbation range of the weights is asymptotically linearly dependent on the radius of the neighborhood. Numerical experiments are proposed to test the asymptotic sharpness of the theoretical results. (C) 2019 Elsevier Inc. All rights reserved.