## 12.2 Natural splines

A natural spline of order k, with knots at $$t_1 <...< t_m$$, is a piecewise polynomial function f such that

• f is polynomial of degree k on each of $$[t_1,t_2],...,[t_{m-1},t_m]$$
• f is a polynomial of degree $$(k-1)/2$$ on $$(-\infty,t_1]$$ and $$[t_m,\infty)$$
• f is continuous and has continuous derivatives of orders 1,.,,, k -1 at its knots $$t_1,..,t_m$$

Note

natural splines are only defined for odd orders k.