Quadrature in Besov spaces on the Euclidean sphere

Let q ≥ 1 be an integer, S denote the unit sphere embedded in the Euclidean space Rq+1, and μq be its Lebesgue surface measure. We establish upper and lower bounds for sup f∈B p,ρ ∣∣∣∣ ∫ Sq fdμq − M ∑ k=1 wkf(xk) ∣∣∣∣ , xk ∈ S , wk ∈ R, k = 1, · · · ,M, where B p,ρ is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of xk and wk that admit exact quadrature for spherical polynomials of a given degree, and satisfy a certain continuity condition; the lower bounds are obtained for the infimum of the above quantity over all choices of xk and wk. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established. ∗The support of the Australian Research Council is gratefully acknowledged. Part of the work was carried out while the author was a guest of the Center for Constructive Approximation at Vanderbilt University. †The research of this author was supported, in part, by grant W911NF-04-1-0339 from the U.S. Army Research Office, grant DMS-0204704, and its continuation grant DMS-0605209 from the National Science Foundation. ‡The support of the Australian Research Council is gratefully acknowledged.