Nearly monotone and nearly convex approximation by smooth splines in \(\mathbb L_p, p>0\) (Q1041623)

The authors give a monotone or convex continuous functions on a finite interval and construct splines of arbitrarily high order having maximum smoothness which are nearly monotone or nearly convex and provide the rate of \(\mathbb L_p\)-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). The authors show how to construct approximating splines of higher orders having maximum smoothness (minimum defect) while preserving their shape properties as well as the approximation orders.