On k-Monotone Approximation by Free Knot Splines

Let SN,r be the (nonlinear) space of free knot splines of degree r − 1 with at most N pieces in [a, b], and let M be the class of all k-monotone functions on (a, b), i.e., those functions f for which the kth divided difference [x0, . . . , xk]f is nonnegative for all choices of (k+1) distinct points x0, . . . , xk in (a, b). In this paper, we solve the problem of shape preserving approximation of k-monotone functions by splines from SN,r in the Lp-metric, i.e., by splines which are constrained to be k-monotone as well. Namely, we prove that the order of such approximation is essentially the same as that by the non-constrained splines. Precisely, it is shown that, for every k, r, N ∈ N, r ≥ k, and any 0 < p ≤ ∞, there exist constants c0 = c0(r, k) and c1 = c1(r, k, p) such that dist(f, Sc0N,r ∩ M )p ≤ c1 dist (f, SN,r)p ∀f ∈ M k . This extends to all k ∈ N results obtained earlier by Leviatan & Shadrin and by Petrov for k ≤ 3.