Piecewise Polynomials

Definition

Suppose that knots are ${\xi }_j,j=1,\dots ,K$. Then, the basis functions for order-M spline are defined as: $\begin{array}{rl}{h}_j(X)& =X^{j-1},j=1,\dots ,M\\ h_{M+l}(X)& =(X-{\xi }_l{)}_{+}^{M-1},l=1,\dots ,K\end{array}$ where $(x{)}_{+}=\{\begin{array}{ll}x& x\ge 0\\ 0& x<0\end{array}$

Examples

For cubic Polynomials with two knots ${\xi }_1,{\xi }_2$, the basis functions are: $h_1(X)=1,h_2(X)=X,h_3(X)=X^2,h_4(X)=X^3,h_5(X)=(X-{\xi }_1{)}_{+}^3,h_6(X)=(X-{\xi }_2{)}_{+}^3$

Facts

Order of piecewise polynomials

• constant: 1
• linear: 2
• quadratic: 3
• cubic: 4