Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials

Abstract

In 2000, Reimer proved that a positive weight quadrature rule on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{{d + 1}}$ has the property of quadrature regularity. Hesse and Sloan used a related property, called Property (R) in their work on estimates of quadrature error on $\mathbb{S}^{d}$. The constants related to Property (R) for a sequence of positive weight quadrature rules on $\mathbb{S}^{d}$ can be estimated by using a variation on Reimer’s bounds on the sum of the quadrature weight within a spherical cap, with Jacobi polynomials of the form $P^{{({1 + d} \mathord{\left/ {\vphantom {{1 + d} 2}} \right. \kern-\nulldelimiterspace} 2,d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2)}}_{t}$, in combination with the Sturm comparison theorem. A recent conjecture on monotonicities of Jacobi polynomials would, if true, provide improved estimates for these constants.