Linear independence of the blending functions of T-splines without multiple knots

Abstract

The purpose of the present paper is to answer the following open question given by Sederberg’s research team: are there T-splines with linearly dependent blending functions that do not have multiple knots? First, the mathematical properties of the blending functions of T-splines without multiple knots are analyzed. Then, the linear independence of them is proved, and meanwhile a necessary condition is deduced for the linear dependence of T-spline blending functions. Finally, the answer to the open question is obtained: there are no T-splines with linearly dependent blending functions that do not have multiple knots.