Optimal lower bounds for cubature error on the sphere S2

We show that the worst-case cubature error E(Q(m); H-s) of an m-point cubature rule Q(m) to r functions in the unit ball of the Sobolev space H-s = H-s (S-2), s > 1, has the lower bound E (Q(m); H-s) >= c(s)m(- s/2), where the constant c(s) is independent of Q and in. This lower bound result is optimal, since we have established in previous work that there exist sequences (Q(m(n)))(n epsilon N) of cubature rules for which E (Q(m(n)); H-s) <= c(s) (m(n))(-s/2) with a constant E, independent of n. The method of proof is constructive: given the cubature rule Qm, we construct explicitly a 'bad' function f(m) epsilon H-s, which is a function for which Q(m)f(m) = 0 and vertical bar vertical bar f(m)vertical bar vertical bar(-1)(Hs)vertical bar integral(2)(S) f(m) (x) d omega (x) >= c(s)m(-s/2). The construction uses results about packings of spherical caps on the sphere. (c) 2005 Elsevier Inc. All rights reserved.