Widths of weighted Sobolev classes on the ball

Abstract

We study the Kolmogorov n -widths d n ( B W p , μ r , L q , μ ) and the linear n -widths δ n ( B W p , μ r , L q , μ ) of weighted Sobolev classes B W p , μ r on the unit ball B d in L q , μ , where L q , μ , 1 ≤ q ≤ ∞ , denotes the weighted L q space of functions on B d with respect to weight ( 1 − | x | 2 ) μ − 1 2 , μ ≥ 0 . Optimal asymptotic orders of d n ( B W p , μ r , L q , μ ) and δ n ( B W p , μ r , L q , μ ) as n → ∞ are obtained for all 1 ≤ p , q ≤ ∞ and μ ≥ 0 .