Legendre-like orthogonal basis for spline space

Abstract

The usual B-spline basis is not orthogonal. In order to resolve the theoretical problem that there is not a well-expressed orthogonal basis in spline space to date, we construct an orthogonal basis for the n-degree spline space in which n is an arbitrary natural number. We extend the traditional Legendre method to spline space and obtain a unified and explicit expression for the orthogonal basis. We first define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions. We also provide some examples of cubic orthogonal splines to demonstrate our process. Finally, the orthogonal basis is applied to the problem of the least-square approximation of curves.