Optimal error bounds for the cubic spline interpolation of lower smooth function. II
For part I see the author in Approximation Theory Appl. 9, No. 4, 46-54 (1993; Zbl 0786.41011). For \(n \in \mathbb N\) and a function \(f \in C^1[0,1]\) let \(s_f\) denote a clamped cubic spline, interpolating \(f\) at \(n+1\) equally spaced nodes \(x_i n\), \(i, \dots, n\). The optimal error bound \[ e_{m,r}(n) \sup_{f \in C^m[0,1]} {\| s_f^{(r)} -f^{(r)} \| _\infty \over \| f^{(m)} \| _\infty} n^{m-r} \] is studied. Using an explicit expression of the reproducing kernel of the interpolating splines, the author finds upper and lower bounds for \(e_{1,0}(n)\) and \(e_{2,2}(n)\), as well as the exact value of \(\sup_n e_{1,0}(n)\) and \(\sup_n e_{2,2}(n)\).
scientific article; zbMATH DE number 1193550
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