Comonotone approximation by splines of piecewise monotone functions

Estimates on the Approximation of 3-monotone Functions by 3-monotone Quadratic Splines

Free-knot Splines Approximation of s-monotone Functions

Abstract. Let I be a finite interval and r, s ∈ N. Given a set M , of functions defined on I, denote by ∆+M the subset of all functions y ∈ M such that the s-difference ∆τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆+W r p , the class of functions x on I with the seminorm ‖x‖Lp ≤ 1, such that ∆τ x ≥ 0, τ > 0. Let Mn(hk) := Pn i=1 cihk(wit − θi) | ci, wi, θi ∈ R , be a single hidden la...

On 3-monotone approximation by piecewise polynomials

Abstract. We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials ...

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 ...

On monotone and convex approximation by splines with free knots

We prove that the degree of shape preserving free knot spline approximation in L p a; b], 0 < p 1 is essentially the same as that of the non-constrained case. This is in sharp contrast to the well known phenomenon we have in shape preserving approximation by splines with equidistant knots and by polynomials. The results obtained are valid both for piecewise polynomials and for smooth splines wi...