Optimal Approximation on

The theory of functions on the two-dimensional sphere was initiated in the 18th century in the works of Laplace and Legendre. The general theory for the d-dimensional sphere was started at the beginning of this century. Due to the technical difficulties involved, the general theory is relatively incomplete, as compared to the theory of functions on the circle. We are introducing a wide range of smooth functions on the d-dimensional sphere and considering asymptotic behavior of n-widths of function sets on spheres. These function sets may have finite or infinite smoothness, the latter including either analytic or entire functions. Let us first recall the definitions of linear and Kolmogorov n-widths that will be studied. Let X be a Banach space, BX :=[x # X : &x& 1] be the unit ball of X, and A be a (convex, compact, centrally symmetric) subset of X. doi:10.1006 jcom.2000.0543, available online at http: www.idealibrary.com on