Notes on lacunary interpolation with splines IV. (0,2) interpolation with splines of degree 6 (Q1112276)

The authors study the following (0,2)-interpolation problem: Given \(\Delta =\{x_ i\}^ n_{i=0}\), \(x_ i=ih\) and the real numbers \(\{f_ i,f_ i''\}^ n_{i=0}\), find S such that \(S(x_ i)=f_ i\), \(S''(x_ i)=f_ i''\), \(i=0,1,...,n\). Using piecewise polynomials of degree 6, they construct a solution S of this problem and show that: if \(f\in C^ 6[0,1]\), then \(| f^{(j)}(x)-S^{(j)}(x)| \leq C^*_{kj}h^{6-j}\omega (f^{(6)};h),\) for \(x_ k\leq x\leq x_{k+1}\), \(0\leq k\leq n-1\), \(0\leq j\leq 6\), where \(C^*_{kj}\) are effectively calculated.