Local quadrature formulas on the sphere

• Authors: H. N. Mhaskar
• Venue: Journal of Complexity
• Year: 2004

Let q ≥ 1 be an integer, Sq be the unit sphere embedded in Rq+1, and µq be the volume element of Sq. For x0 ∈ Sq, and α ∈ (0, π), let Sαq(x0) denote the cap {ξ ∈ Sq : x0 ċ ξ ≥ cos α}. We prove that for any integer m ≥ 1, there exists a positive constant c=c(q, m), independent of α, with the following property. Given an arbitrary set C of points in Sαq(x0), satisfying the mesh norm condition maxξ∈Sαq(x0) minζ∈C dist(ξ,ζ) ≤ cα, there exist nonnegative weights wξ, ξ ∈ C, such that ∫Sαq(x0) P(ζ)dµq(ζ) = Σξ ∈C wξP(ξ) for every spherical polynomial P of degree at most m. Similar quadrature formulas are also proved for spherical bands.