Estimates of Kolmogorov's and linear n-widths of Sobolev's classes on compact globally symmetric spaces of rank 1 (i.e. on S-d, P-d(R), P-d(C), P-d(H), P-16(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. New estimates for the (p,q)-norms of multiplier operators Lambda = {lambda(k)}(kis an element ofN) are given. We apply our results to get sharp orders of best polynomial approximation and n-widths. (C) 2002 Elsevier Inc. All rights reserved.2022307326