Natural Cubic and Bicubic Spline Interpolation

Abstract

An alternate proof is presented of Kershaw’s result that the $L_\infty $-norm of the error in natural spline interpolation to a function $f \in C^4 [a,b]$ is $O(h^4 )$ in a closed subinterval which is asymptotic to $[a,b]$ as $h \to 0$. The case $f \in C^m [a,b],m = 2$ or 3, is also considered. These univariate results are then used to investigate natural bicubic spline interpolation over a rectangle $\mathcal{R}$. For $f \in C^m [\mathcal{R}]$, the $L_\infty $-norm of the error is shown to be $O(h^{m - 2} )$ throughout $\mathcal{R}$ and $O(h^m )$ in a closed subrectangle which is asymptotic to $\mathcal{R}$ as $h \to 0$. Arbitrary sequences of partitions are considered throughout.