On k-Monotone Approximation by Free Knot Splines

Abstract

Let $\mathcal{S}_{N,r}$ be the (nonlinear) space of free knot splines of degree r-1 with at most N pieces in [a,b], and let $\mathcal{M}^k$ 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 $\mathcal{S}_{N,r}$ in the $\mathbb{L}_p$-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 nonconstrained splines. Precisely, it is shown that, for every $k,r,N\in\mathbb{N}$, $r\geq k$, and any $0<p\leq\infty$, there exist constants c0 = c0 (r,k) and c1 = c1 (r,k,p) such that $$ \mathop{\rm dist}(f, \mathcal{S}_{c_0 N, r}\cap\M^k)_p \leq c_1 \mathop{\rm dist}(f, \mathcal{S}_{N,r} )_p \quad \forall f \in \M^k . $$ This extends to all $k...