Explicit representations of changeable degree spline basis functions (Q1758401)

The authors study changeable degree (CD) spline basis functions and present explicit representations for them. The method for explicit representations is based on four steps: Firstly, a family of truncated power functions \(F_{i,D}\), \(i\in \mathbb{Z}\), is defined. Secondly, it is shown that if a function has a local support interval and fulfills some continuity conditions on this interval, then it is unique up to a multiplicative constant. Thirdly, for each CD-spline basis function \(N_{i,D}\), a determinant related to the truncated power functions \(F_{i,D}\) is used to obtain a function that has the same support interval and continuity properties as \(N_{i,D}\). Thus, \(N_{i,D}\) is expressible as this determinant times a constant. Fourthly, this constant is obtained from the normalized property of CD-spline basis functions. Some results based on an explicit representation procedure are stated and extensions to other spline systems are indicated.