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Error estimates for the cardinal spline interpolation. (English) Zbl 1171.41005
This paper is devoted to a study of cardinal spline interpolation in the Sobolev space ${V}^{m,\infty }\left(ℝ\right)$ of functions with a bounded $m$th derivative on the real line. Specifically, for the interpolation using cardinal splines of degree $m-1$ with knots $ih$ and interpolation nodes $\left(i+1/2\right)h$, $i\in ℤ$, with some $h>0$, it is well known that the interpolation error is uniformly bounded by ${{\Phi }}_{m+1}{\pi }^{-m}{h}^{m}{\parallel f\parallel }_{\infty }$, where ${{\Phi }}_{m+1}$ is the $\left(m+1\right)$st Favard constant, if $f\in {V}^{m,\infty }\left(ℝ\right)$ and, additionally, $f$ is 1-periodic. The paper under review demonstrates that the same error bound holds for all $f\in {V}^{m,\infty }\left(ℝ\right)$, no matter whether they are periodic or not. Moreover it is shown in a constructive way that this error bound is unimprovable. Additional results presented in the paper include, among others, bounds for the difference between derivatives of the function to be interpolated and the corresponding derivatives of the interpolating splines.
##### MSC:
 41A15 Spline approximation 41A05 Interpolation (approximations and expansions) 41A44 Best constants (approximations and expansions)