Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, r,n∈N, s∈N0 and 1⩽p⩽∞. Given a set M, of functions defined on I, denote by the subset of all functions y∈M such that the s-difference is nonnegative on I, ∀τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm ∥x(r)∥Lp⩽1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in Lq for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n-2.