Quadratic and cubic spline interpolation (Q788928)

Let (k,m) denote the class of k degree interpolating polynomial splines whose interpolation conditions are given by mth derivative values. Here we study the (2,0) and (3,1) problems. For \(-\infty<a<b<+\infty\) and for any positive integer \(n\geq 2\), let \(\Delta:a=x_ 0<x_ 1<...<x_ n=b\) denote a partition of [a,b] with knots \(x_ i\) and steps \(h_ i=x_{i+1}-x_ i\). Let S\(p(\Delta\),k) denote the class of k degree polynomial splines on \(\Delta\). (2,0) interpolation problem: Find s(x)\(\in Sp(\Delta,2)\) such that \(s(x_ i)=f_ i (i=0,1,...,n)\), \(s'(x_ j)=f_ j\!' (j=0\) or n). (3,1) interpolation problem: Find s(x)\(\in Sp(\Delta,3)\) such that \(s'(x_ i)=f_ i\!' (i=0,1,...,n)\), \(s(x_ j)=f_ j (j=0\) or n), \(s''(x_ j)=f_ j\!'' (j=0\) or n). The interpolation problems above are well-defined. Define the functionals \[ J[f^{(k-1)}]=\sum^{n-1}_{i=0}J_ i[f^{(k-1)}]\quad(k=2,3), \] \[ J_ i[f^{(k-1)}]=\int^{x_{i+1}}_{x_ i}[f^{(k-1)}(x)+f^{(k- 1)}(x_ i+x_{i+1}-x)]^ 2dx,\quad(i=0,1,...,n-1). \] We have the first and second integration relationships and the minimization properties. Let \(\| f^{(j)}\| =\max_{x\in [a,b]}| f^{(j)}(x)|, V^ b_ a(f^{(j)})=\int^{b}_{a}| df^{(j)}|, M_ j=\| f^{(j)}\| +V^ b_ a(f^{(j)}), h=\max_{0\leq i\leq n-1}h_ i\), \(\max h_ i/\min h_ i\leq \beta_ 0, (\sum^{n-1}_{j=1}| h_ j-h_{j+1}|)/h\leq \beta_ 1\), \(\max_{0\leq m\leq n-1}(\max_{m<1}(\sum^{i}_{j=m+1}| h_ j- h_{j+1}|)/h_ m\), \(\max_{i<m}(\sum^{m}_{j=i+1}| h_ j- h_{j+1}|)/h_ m\leq \beta_ 2, R(f,x)=f(x)-s_ f(x),\) where \(s_ f(x)\in Sp(\Delta,k)\) is the solution of the (k,k-2) interpolation problem for f(x). We have the following error estimates. If \(f(x)\in C^ 3[a,b]\) with f'''(x) of bounded variation, then for the solution of the (2,0) problem and its derivatives we have \(\| R^{(j)}(f,x)\| \leq c_ jM_ 3h^{3-j} (j=0,1,2)\), where \(c_ 0=(4+3\beta_ 1)/36,\quad c_ 1=(4+3\beta_ 1)/18,\quad c_ 2=(5+\beta_ 0+2\beta_ 2)/6.\) If \(f(x)\in C^ 4[a,b]\) with \(f^{(4)}\) of bounded variation, then for the solution of the (3,1) problem and its derivatives we have \[ \| R(f,x)\| \leq c_ 0(b-a)M_ 4h^ 3,\quad \| R^{(j+1)}(f,x)\| \leq c_ jM_ 4h^{3-j}\quad(j=0,1,2), \] where \(c_ 0,c_ 1,c_ 2\) are the same as above.