mn header will be provided by the publisher Estimates of n–Widths of Besov Classes on Two-Point Homogeneous Man- ifolds A. K. Kushpel ∗1 , J. Levesley ∗∗2 , and S. A. Tozoni ∗∗∗3 1 Department of Mathematics, Physics and Computer Science, Ryerson University, Toronto, ON M5B 2K3, Canada. 2 Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, UK. 3 Instituto de Matematica, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970 Campinas, SP, Brazil. Received 15 November 2003, revised 30 November 2003, accepted 2 December 2003 Published online 3 December 2003 Key words n–width, two point homogeneous manifold, Besov space. MSC (2000) 41A46, 42B15 Estimates of Kolmogorov and linear n-widths of Besov classes on compact globally symmetric spaces of rank 1 (i.e. on S d , P d (IR), P d ( l C), P d (IH), P 16 (Cay) ) are established. It is shown that these estimates have sharp orders in different important cases. A new characterisation of Besov spaces is also given. Copyright line will be provided by the publisher 1 Introduction In the present paper we investigate the asymptotically optimal approximation of Besov classes on compact glob- ally symmetric spaces of rank 1 (two-point homogeneous spaces) S d , P d (IR), P d ( l C), P d (IH), P 16 (Cay). In what follows, optimal approximation will be interpreted in the sense of Kolmogorov and linear n-widths. Estimates for Kolmogorov n-widths of Besov classes on bounded regions of Euclidean spaces can be found in [21]. The spaces of Besov type on manifolds and their equivalent characterisations have been investigated in different articles (see e.g. [22, 23, 16, 15, 5, 17]). There are various approaches to the definition of smoothness via harmonic analysis. The basic theorem in this range of problems is the well known analog of the Littlewood-Paley theorem [12] for trigonometric series, on compact globally symmetric spaces of rank 1 by Bonami and Clerc [2]. We introduce the Besov spaces decomposing a smooth function f into a series relative to spherical harmonics and using zonal polynomials K n (z) which are natural generalizations of the de la Vall´ ee Poussin polynomials on S 1 . We prove that the Besov spaces are real interpolation of two Sobolev spaces. Our definition of Besov space is new even for the sphere S d , d ≥ 2. We use sharp orders of Kolmogorov n-widths of Sobolev classes from [3] and [11], and interpolation tech- niques by Triebel [21] to prove asymptotic estimates for Kolmogorov and linear n-widths of Besov classes on two-point homogeneous spaces. Suppose that A is a convex, compact, centrally symmetric subset of a Banach space X with unit ball B. The linear n–width of A in X is defined by δ n (A, X) := δ n (A, B) := inf P n sup f ∈A ‖f − P n f ‖, where P n varies over all linear operators of rank at most n that map X into itself. The Kolmogorov n–width of A in X is defined by d n (A, X) := d n (A, B) := inf Xn sup f ∈A inf g∈Xn ‖f − g‖, * Corresponding author: e-mail: akushpel@ryerson.ca ** Corresponding author: e-mail: jl1@mcs.le.ac.uk *** Corresponding author: e-mail: tozoni@ime.unicamp.br Copyright line will be provided by the publisher