L _p -Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

Abstract.

The B-spline representation for divided differences is used, for the first time, to provide L _p -bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities.

The major result is the inequality $|f(x) - H_{\Theta} f(x)| \leq \frac{n^{1/q}}{n!} \frac{|\omega_{\Theta}(x)|}{(\mathop{\rm diam}\nolimits \{ x , \Theta \} )^{1/q}} \| D^n f\|_q,$ where H_Θ f is the Hermite interpolant to f at the multiset of n points Θ, $\omega_{\Theta} (x) := \prod_{\theta \in \Theta} (x-\theta),$ and $\mathop{\rm diam}\nolimits \{ x, \Theta \}$ is the diameter of $\{ x, \Theta \}$ . This inequality significantly improves upon Beesack's inequality, on which almost all the bounds given over the last 30 years have been based.