Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I , τ>0 . Further, denote by , 0≤α<∞ , the classes of functions x on I with the seminorm ||x ^(r) ρ ^α ||_ L _p ≤ 1 , such that Δ ^s _τ x≥ 0 , τ>0 . For s=0,1,2 , we obtain two-sided estimates of the shape-preserving widths where M ⁿ is the set of all linear manifolds M ⁿ in L _q , such that dim M ⁿ ≤ n , and satisfying .