Shape-preserving widths of weighted Sobolev-type classes of positive, monotone, and convex functions on a finite interval

Let I be a finite interval, r is an element of N and p(t) = dist(t, partial derivativeI), t is an element of I. Denote by Delta(+)(s) Lq the subset of all functions y is an element of L-q such that the s-difference Delta(tau)(s) y(t) is nonnegative on I, For All tau > 0. Further, denote by Delta(+)(s) W-p.alpha(r), 0 less than or equal to alpha < infinity, the classes of functions x on I with the seminorm parallel tox((r)) rho(alpha)parallel to L-p less than or equal to 1, such that Delta(tau)(s) x greater than or equal to 0, tau > 0. For s = 0, 1, 2, we obtain two-sided estimates of the shape-preserving widths d(n)(Delta(+)(s) W-p.alpha(r), Delta(+)(s) L-q)L-q := inf(Mn is an element of Mn) sup(x is an element of Delta+s Wp.alphar) inf(y is an element of Mn boolean AND Delta+s Lq) parallel tox-yparallel to(Lq), where M-n is the set of all linear manifolds M-n in L-q, such that dim M-n less than or equal to n, and satisfying M-n boolean AND Delta(+)(s) L-q not equal 0.