Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials

In 2000, Reimer proved that a positive weight quadrature rule on the unit sphere Sd ⊂ Rd+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 Sd . Using a variation on Reimer’s bounds on the sum of the quadrature weight within a spherical cap, and using Jacobi polynomials of the form P (1+d/2,d/2) t , as well as the Sturm comparison theorem, the constants related Property (R) can be estimated. A recent conjecture on monotonicities of Jacobi polynomials would, if true, provide improved estimates for these constants.

• Type: Journal
• Year: 2007
• Where: NA
• Authors: Paul C. Leopardi
• Keywords: Jacobi Polynomials, NA 2007, Positive Weight Quadrature, Quadrature Regularity