Complexity of numerical integration over spherical caps in a Sobolev space setting

Let r=2, let Sr be the unit sphere in Rr+1, and let View the MathML source be the spherical cap with center View the MathML source and radius ??(0,p]. Let Hs(Sr) be the Sobolev (Hilbert) space of order s of functions on the sphere Sr, and let Qm be a rule for numerical integration over View the MathML source with m nodes in View the MathML source. Then the worst-case error of the rule Qm in Hs(Sr), with s>r/2, is bounded below by cr,s,?m-s/r. The worst-case error in Hs(Sr) of any rule Qm(n) that has m(n) nodes in View the MathML source, positive weights, and is exact for all spherical polynomials of degree =n is bounded above by View the MathML source. If positive weight rules Qm(n) with m(n) nodes in View the MathML source and polynomial degree of exactness n have m(n)~nr nodes, then the worst-case error is bounded above by View the MathML source, giving the same order m-s/r as in the lower bound. Thus the complexity in Hs(Sr) of numerical integration over View the MathML source with m nodes is of the order m-s/r. The constants cr,s,? and View the MathML source in the lower and upper bounds do not depend in the same way on the area View the MathML source of the cap. A possible explanation for this discrepancy in the behavior of the constants is given. We also explain how the lower and upper bounds on the worst-case error in a Sobolev space setting can be extended to numerical integration over a general non-empty closed and connected measurable subset O of Sr that is the closure of an open set