Feng Dai

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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 d_n(B_p^r,L^q) and linear n-widths δ_n(B_p^r,L^q), (1 ≤ q ≤∞) of Sobolev’s classes B_p^r, (r > 0, 1 ≤ p ≤∞) on the unit sphere S^d-1 of the d-dimensional Euclidean space R^d. For part of (p,q)

[1,∞] × [1,∞], sharp orders of d_n(B_p^r,L^q) or δ_n(B_p^r,L^q) were previously known. In our work, we obtained the sharp orders of d_n(B_p^r,L^q) and δ_n(B_p^r,L^q) 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 S^d-1. Our work also reveals a close relation between positive cubature formulas and MZ inequalities on S^d-1.

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