Let x0

(a) Construct and graph the natural cardinal splines corresponding to the nodes x0 = 0, x1 = 1, x2 = 2, and x3 =3.
(b) Prove that the natural spline that interpolates
the data yo,... ,yn can be uniquely written as a linear combination u(x) = yoCo(x) + y1C1y +......+ YnCn(x) of the cardinal splines.
(c) Explain why the space of natural splines on n+1 nodes is a vector space of dimension n+1.
(d) Discuss briefly what modifications are required to adapt this method to periodic and to clamped splines.

Yo = 0, y, = 1, yj+1 = 0, Y = 0, Y-1 = 0, Ya = 0.