Multivariate approximating averages Original Research Article

Averages on the sphere about x in Rd and on the rim of the cap about x in Sd−1 and their iterates are shown to be smoother than f. Furthermore, their approximating properties satisfy a strong converse inequality of type A when dealing with multivariate approximation (d⩾2 in case of Rd and d⩾3 in case of Sd−1). These results are in contrast to the classical results on R or T for which the situation is completely different.