Basis For Spline Space Research Articles

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.

Legendre-like orthogonal basis for spline space

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.

On the foundation of bases of spline spaces

The concept of foundation is a sharper concept than support which allows us to recognize spline bases suitable for design and computational purposes. We present new results on least founded bases of univariate spaces. In the case of tensor product spaces, we find under very general conditions bases of minimally founded functions although we show that, in contrast to the univariate case, they are no least founded bases. We also analyze related questions in a space of splines defined on angular regions of the plane.