Nearly monotone and nearly convex approximation by smooth splines in Lp, p>0

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are “nearly monotone” or “nearly convex” and provide the rate of Lp-approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or Chebyshev knots). It is known that these estimates are impossible in terms of higher moduli and are no longer true for “purely monotone” and “purely convex” spline approximation.