Natural Cubic Splines

Definition

Suppose that knots are ${\xi }_j,j=1,\dots ,K$. Then, the basis functions for natural cubic Spline are defined as: $N_1(X)=1,N_2(X)=X,N_{k+2}(X)=d_k(X)-d_{K-1}(X)$ where $d_k(X)=\frac{(X-{\xi }_k{)}_{+}^3-(X-{\xi }_K{)}_{+}^3}{{\xi }_K-{\xi }_k}$