Kolmogorov and linear widths of weighted Besov classes

We study the Kolmogorov m-widths dm(Bα s (Lτ,μ),Lp,μ) and the linear m-widths δm(Bα s (Lτ,μ),Lp,μ) of the weighted Besov classes Bα s (Lτ,μ) on [−1, 1], where Lq,μ, 1 q ∞, denotes the Lq space on [−1, 1] with respect to the measure (1− t2)μ−1/2 dt, μ>0. Optimal asymptotic orders of dm(Bα s (Lτ,μ), Lp,μ) and δm(Bα s (Lτ,μ),Lp,μ) as m→∞are obtained for all 1 p, τ ∞. It turns out that in many cases, the orders of dm(Bα s (Lτ,μ),Lp,μ) are significantly smaller than the corresponding orders of the best m-term approximation by ultraspherical polynomials, which is somewhat surprising.  2005 Elsevier Inc. All rights reserved.