Quadrature in Besov spaces on the Euclidean sphere

Let q⩾1q⩾1 be an integer, SqSq denote the unit sphere embedded in the Euclidean space Rq+1Rq+1, and μqμq be its Lebesgue surface measure. We establish upper and lower bounds forsupf∈Bp,ργ∫Sqfdμq-∑k=1Mwkf(xk),xk∈Sq,wk∈R,k=1,…,M,where Bp,ργ is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of xkxk and wkwk 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 xkxk and wkwk. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established.