# Extremising the $L_p$-norm of a monic polynomial with roots in a given interval and Hermite interpolation

## Abstract:

Let $\Theta$ be a multiset of $n$ points in $[a,b]$, and
$$\omega_\Theta:=\prod_{\theta\in\Theta}(\cdot-\theta).$$
In this paper we investigate the extrema of $\Theta\mapsto\norm{\omega_\Theta}_p$. Consequences of the results we obtain include: $L_p$-bounds for Hermite interpolation, error estimates for Gauss quadrature formul{\ae} with multiple nodes, and certain quantitative statements about good and best approximation by polynomials of fixed degree.

Keywords: Hermite interpolation, B-spline, Green's function, Beesack's inequality, Wirtinger inequality

Math Review Classification: xxx (primary), xxx (secondary)

Length: 10 pages

Comment: Written in TeX, contains 1 figure. Supported by the Chebyshev professorship of Carl de Boor. See Project Hermite for related work.

Last updated: 25 March 1996

Status: This article will be reincarnated at some date in the near future (to deal with the derivatives of such polynomials, etc.)