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

Abstract

Let r ≥ 2 , let S r be the unit sphere in R r + 1 , and let C ( z ; γ ) : = { x ∈ S r : x ⋅ z ≥ cos γ } be the spherical cap with center z ∈ S r and radius γ ∈ ( 0 , π ] . 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 ; γ ) with m nodes in C ( z ; γ ) . 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 , γ 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 ; γ ) , positive weights, and is exact for all spherical polynomials of degree ≤ n is bounded above by c ̃ r , s , γ n − s . If positive weight rules Q m ( n ) with m ( n ) nodes in C ( z ; γ ) and polynomial degree of exactness n have m ( n ) ∼ n r nodes, then the worst-case error is bounded above by c ˆ r , s , γ ( m ( n ) ) − s / r , giving the same order m − s / r as in the lower bound. Thus the complexity in H s ( S r ) of numerical integration over C ( z ; γ ) with m nodes is of the order m − s / r . The constants c r , s , γ and c ˆ r , s , γ in the lower and upper bounds do not depend in the same way on the area | C ( z ; γ ) | ∼ γ 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 Ω of S r that is the closure of an open set.