Local quadrature formulas on the sphere

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 Sα(x0) denote the cap {ξ ∈ Sq : x0 · ξ ≥ cosα}. 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 Sα(x0), satisftying the mesh norm condition max ξ∈Sα(x0) min ζ∈C dist (ξ, ζ) ≤ cα, there exist nonnegative weights wξ, ξ ∈ C, such that ∫ Sα(x0) P (ζ)dμq(ζ) = ∑ ξ∈C wξP (ξ) for every spherical polynomial P of degree at most m. Similar quadrature formulas are also proved for spherical bands.