Numerical integration with polynomial exactness over a spherical cap

Abstract

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere $\mathbb{S}^2$ , we discuss tensor product rules with n 2/2?+?O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ??n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on $\mathbb{S}^2$ . For arbitrary d???2, this strategy is extended to provide rules for numerical integration over spherical caps on $\mathbb{S}^d$ that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ??n. We also show that positive weight rules for numerical integration over spherical caps on $\mathbb{S}^d$ that are exact for all spherical polynomials of degree ??n have at least O(n d ) nodes and possess a certain regularity property.