Semilocal smoothing splines (Q460527)

The author considers periodic and nonperiodic semi-local smoothing splines, called \(S\)-splines of class \(C^p\), generated by polynomials of degree \(n\). In the construction of such splines the first \(p+1\) coefficients of each underlying polynomial are determined by the values of the preceding polynomial and its first \(p\) derivatives at the knot points, whereas the remaining \(n-p\) coefficients for the higher order derivatives are determined by a least-square approximation. The author derives a linear system whose matrix has block structure for the polynomial coefficients that define the \(S\)-spline. In addition, existence and uniqueness theorems are proved. Examples of \(S\)-splines are given.