An adaptive B-spline representation of piecewise polynomial functions for multilevel approximation

Abstract

Let I be some real interval endowed with an arbitrary partition τ. The aim of this work is to establish a new B-spline representation of the piecewise polynomial functions (p.p.f.) defined on I which can be used for a multilevel approximation. First, we show that any p.p.f. S can be written in terms of B-splines having τ as sequence of simple knots and the same smoothness order as S. This new family of B-splines has interesting properties similar to those of the well known B-splines. In order to illustrate the interest of this representation, we establish two methods which allow to approximate or project a given p.p.f. S in spaces with high smoothness order. At the end of this paper, we use this representation for constructing quasi-interpolants based on these methods.

Keywords:

• B-splines
• p.p.f.
• smoothing
• quasi-interpolants

2000 AMS Subject Classifications: :

• 41A05
• 41A15
• 65D05
• 65D07
• 65D10