Natural and smoothing quadratic spline. (An elementary approach) (Q808149)

The authors study the problem of constructing a parabolic spline s on [a,b] with knots \(\{x_ i\}^ n_ 1\), \(a=x_ 0<x_ 1<...<x_{n+1}=b\), satisfying the interpolation conditions (1) \(\int^{x_{i+1}}_{x_ i}s(t)dt=y_ i,\) \(i=0,1,...,n\). They show that the unique solution s (under certain boundary conditions) minimizes the integral \(\int^{b}_{a}[s'(t)]^ 2dt\) over the set of all interpolants from \(W^ 1_ 2[a,b].\) Reviewer's remark: Set \(p'(t)=s(t)\), \(p(a)=0\). Then the problem (1) is equivalent to the cubic spline interpolation problem \(p(x_ i)=f_ i,\) \(i=0,1,...,n+1\), with \(f_ 0=0\), \(f_ i:=y_ 0+y_ 1+...+y_{i-1}\), \(i=1,...,n+1\), and many of the results in this paper follow from the known theorems about natural spline interpolation.