Nearly monotone spline approximation in $\mathbb{L}_p$

It is shown that the rate of $\mathbb{L}_p$-approximation of a non-decreasing function in $\mathbb{L}_p$, $0<p<\infty$, by ``nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for ``purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.