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Feng Dai

University of Alberta (Canada)

Kolmogorov and linear n-widths for the Sobolev classes on the unit sphere.

This talk is a joint work with Gavin Brown, conducted when I was a student at the University of Sydney. The main purpose of our research is to estimate Kolmogorov 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.