Approximation by shape preserving interpolation splines

In this paper we present a shape preserving method of interpolation for scattered data defined in the form of some constraints such as convexity, monotonicity and positivity. We define a k-convex interpolation spline function in a Sobolev space, by minimizing a semi-norm of order k + 1. and we discretize it in the space of piecewise polynomial spline functions. The shape preserving condition that we consider here is the positivity of the derivative function of order k. We present an algorithm to compute the resulting function and we show its convergence. Some convergence theorems are established. The error is of order o(1/N)(k+1). where N is the number of the Lagrangian data. Finally we analyze some numerical and graphical examples. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.