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

AbstractLet r≥2, let Sr be the unit sphere in Rr+1, and let C(z;γ):={x∈Sr:x⋅z≥cosγ} be the spherical cap with center z∈Sr and radius γ∈(0,π]. 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 C(z;γ) with m nodes in C(z;γ). Then the worst-case error of the rule Qm in Hs(Sr), with s>r/2, is bounded below by cr,s,γm−s/r. The worst-case error in Hs(Sr) of any rule Qm(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 c̃r,s,γn−s.If positive weight rules Qm(n) with m(n) nodes in C(z;γ) and polynomial degree of exactness n have m(n)∼nr nodes, then the worst-case error is bounded above by cˆr,s,γ(m(n))−s/r, giving the same order m−s/r as in the lower bound. Thus the complexity in Hs(Sr) of numerical integration over C(z;γ) with m nodes is of the order m−s/r. The constants cr,s,γ and cˆr,s,γ in the lower and upper bounds do not depend in the same way on the area |C(z;γ)|∼γr 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 Sr that is the closure of an open set