Impact of the shape of functions on the orders of piecewise polynomial and rational approximation

Let Δs+ be the set of functions x:I→R on a finite interval I such that the divided differences [x;t0,...,ts] of order s element of N of these functions are non-negative for all systems of s+1 distinct points t0,...,ts element of I. Let Σr,n={σr,n} be the set of piecewise polynomial splines σr,n of order r with n-1 free knots, and Rn={ρn} the set of rational functions ρn=π-circumflexn/π-dashn, where π-circumflexn and π-dashn are polynomials of order n. For the classes Δs+Bp:=Δs+ intersection Bp, where Bp is the unit ball in Lp, the precise orders E(Δ+sBp,Σr,n)Lq asymp n-min{r,s} and E(Δ+sBp,Rn)Lq asymp n-s of the best approximations in the Lq metrics are found for 1≤q<p≤∞.