Worst-case errors in a Sobolev space setting for cubature over the sphere S2

This paper studies the problem of numerical integration over the unit sphere S-2 subset of R-3 for functions in the Sobolev space H-3/2(S-2). We consider sequences Q(m(n)), n is an element of N, of cubature (or numerical integration) rules, where Q(m(n)) is assumed to integrate exactly all (spherical) polynomials of degree <= n, and to use m = m(n) values of f. The cubature weights of all rules Q(m(n)) are assumed to be positive, or alternatively the sequence Q(m(n)), n is an element of N, is assumed to have a certain local regularity property which involves the weights and the points of the rules Q(m(n)), n is an element of N. Under these conditions it is shown that the worst-case (cubature) error, denoted by E-3/2(Q(m(n))), for all functions in the unit ball of the Hilbert space H-3/2(S-2) satisfies the estimate E-3/2(Q(m(n))) <= cn(-3/2), where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Q(m(n)), n is an element of N, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Q(m(n)), n is an element of N. Examples of cubature rules that satisfy the assumptions are discussed.