Nearly monotone spline approximation in 𝕃_{𝕡}

Abstract

It is shown that the rate of L p \mathbb {L}_p -approximation of a non-decreasing function in L p \mathbb {L}_p , 0 > 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.