Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris


Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_982
Splines are an important class of mathematical functions used for approximation. A spline is a piecewise polynomial function that is commonly described as being “as smooth as it can be without reducing to a polynomial” (de Boor, 1978, p. 125). For example, the cubic spline shown as the solid line in Figure 1 is composed of individual cubic polynomials, each defined between two adjacent data points, such that the function values and first and second derivatives of adjoining polynomial pieces are the same. In general, a function defined on an interval [a, b] is defined as a polynomial spline of degree k, having knots t1, ..., tr, if the following three conditions hold: (i) a < t1 < ⃛ < tr < b, so the knots t1, ..., tr partition the interval [a, b] into r + 1 smaller subintervals, (ii) on each subinterval [ti, ti+1], the spline is given by a polynomial function of at most degree k, and (iii) the spline and its derivatives up to order k − 1 are all continuous on [a, b]. The definition of...
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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Worcester Polytechnic InstituteWorcesterUSA