Complexity of numerical integration over spherical caps in a Sobolev space setting
Let r >= 2, let S(r) be the unit sphere in R(r+1), and let C(z; gamma) := {x is an element of S(r) : x.z >= cos gamma} be the spherical cap with center Z is an element of S(r) and radius gamma is an element of (0, pi]. Let H(S)(S(r)) be the Sobolev (Hilbert) space of order s of functions on the sphere S(r), and let Q(m) be a rule for numerical integration over C(z; gamma) with m nodes in C(z; gamma). Then the worst-case error of the rule Q(m) in H(s) (S(r)), with s > r/2, is bounded below by c(r.s,gamma) m(-s/r). The worst-case error in H(s) (S(r)) of any rule Q(m,(n)) that has m(n) nodes in C(z; gamma), positive weights, and is exact for all spherical polynomials of degree <= n is bounded above by (C) over tilde (r,s,gamma)n(-s). If positive weight rules Q(m(n)) with m(n) nodes in C(z; gamma) and polynomial degree of exactness n have m(n) similar to n(r) nodes, then the worst-case error is bounded above by (C) over cap (r,s,gamma) (m(n))(-s/r), giving the same order m(-s/r) as in the lower bound. Thus the complexity in H(r)(S(r)) of numerical integration over C(z; gamma) with m nodes is of the order m(-s/r). The constants c(r,s,gamma) and (C) over cap (r,s,gamma) in the lower and upper bounds do not depend in the same way on the area vertical bar C(z; gamma)vertical bar similar to gamma(r) 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 Omega of S(r) that is the closure of an open set. (C) 2010 Elsevier Inc. All rights reserved.
complexity numerical integration spherical cap sobolev space worst-case error error estimates