Optimal smoothing and interpolating splines with constraints Hiroyuki Kano a,⇑ , Hiroyuki Fujioka b , Clyde F. Martin c a Division of Science, Tokyo Denki University, Saitama 350-0394, Japan b Department of System Management, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan c Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA article info Keywords: B-splines Optimal smoothing splines Optimal interpolating splines Equality/inequality constraint Quadratic programming abstract This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employ- ing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to qua- dratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approxima- tion of discontinuous functions, and trajectory planning. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Splines are an essential part of the applied mathematics toolbox [1] and for the last several years it has been known that they are closely related to a series of linear control problems [2–5]. In this paper we extend this connection between optimal control and splines by demonstrating that there is a large class of spline problems that can be solved using quadratic pro- gramming techniques. In many applications traditional splines are not sufficient. Often there are constraints imposed on the spline curve because of the nature of the data. For example, if splines are being used to chart the height of a child over a period of years one looses considerable credibility with the pediatrician if the curve is not monotone increasing. Linear con- straints are easy to impose and can systematically be added to the construction of spline curve [6]. Inequality constraints at isolated points have been imposed on splines [2,3], but the construction of the spline then becomes a quadratic program- ming problem. It is interesting to note that although quadratic programs are known to be sometimes non-convergent that no spline construction with pointwise constraints has ever failed to converge. Inequality constraints over intervals are much more difficult. In [7] monotone smoothing splines were constructed but other types of inequality constraints were not at- tempted. In this paper we show that by using B-splines inequality constraints of all forms can be systematically added and the construction of the spline curves reduces to quadratic programming problems. We also show that by using B-splines the Gibbs phenomena can be controlled. This is in sharp contract to classical cubic splines [8]. This paper is organized as follows. In Section 2, we briefly review B-splines and design methods of optimal splines. Then in Section 3, we show that various types of constraints on splines may be incorporated as quadratic programming problems. We examine the performances of the proposed method by numerical examples in Section 4. Concluding remarks are given in Section 5. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.06.067 ⇑ Corresponding author. E-mail addresses: kano@mail.dendai.ac.jp (H. Kano), fujioka@fit.ac.jp (H. Fujioka), clyde.f.martin@ttu.edu (C.F. Martin). Applied Mathematics and Computation 218 (2011) 1831–1844 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc