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 \(t_1,..,t_m\)


natural splines are only defined for odd orders k.