Expansions, inequalities and approximations.
On the occasion of Gavin Brown’s 65th Birthday
University of Sydney, Australia, 5-6 March 2007
n-widths dn(Bpr,Lq) and linear n-widths δn(Bpr,Lq), (1 ≤ q ≤∞) of Sobolev’s classes
Bpr, (r > 0, 1 ≤ p ≤∞) on the unit sphere Sd-1 of the d-dimensional Euclidean space Rd. For part of
(p,q) [1,∞] × [1,∞], sharp orders of dn(Bpr,Lq) or δn(Bpr,Lq) were previously known. In our work, we
obtained the sharp orders of dn(Bpr,Lq) and δn(Bpr,Lq) for all the remaining cases of (p,q). Our proof is based
on positive cubature formulas and Marcinkiewicz-Zygmund (MZ) inequalities for the spherical polynomials on
Sd-1. Our work also reveals a close relation between positive cubature formulas and MZ inequalities on Sd-1.