BSplines
As a basis for the spline space of all polynomial spline functions of degree d on a knot vector {t_{i}}, we use Bspline functions. A Bspline function is a polynomial spline function which is zero except on a small intervall [t_{j, }t_{j+d+1}]. The number of nonzero Bsplines in a particular interval I = [t_{k, }t_{k+1}] is d+1, and their polynomial part on I form a basis of all ddegree polynomial functions on I. Therefore, when evaluating a spline function for a particular parameter value t, it suffices to evaluate the d+1 Bsplines that are nonzero in t. The evaluation of Bsplines is numerically stable and efficient, which is a major reason for why splines are so widely used in applied mathematics.

 