## L_p-error bounds for Hermite interpolation and the associated Wirtinger inequalities

### BibTeX

@MISC{Waldron_l_p-errorbounds,

author = {Shayne Waldron},

title = {L_p-error bounds for Hermite interpolation and the associated Wirtinger inequalities},

year = {}

}

### OpenURL

### 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 (cf. [FMP91:p66]). The major result is the inequality jf(x) \Gamma H \Theta f(x)j n 1=q n! j! \Theta (x)j (diamfx; \Thetag) 1=q kD n fk q ; where H \Theta f is the Hermite interpolant to f at the multiset of n points \Theta, ! \Theta (x) := Y `2\Theta (x \Gamma `); and diamfx; \Thetag is the diameter of fx; \Thetag. This inequality significantly improves upon `Beesack's inequality' (cf. [Be62]), on which almost all the bounds given over the last 30 years have been based. AMS (MOS) Subject Classifications: primary 26D10, 41A05, 41A80; secondary 41A10, 41A44 Key Words: Hermite interpolation, B-spline, Green's function, Beesack's inequality, Wirti...